---
abstract: 'Making use of Murakami’s classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms of meromorphic functions on $S^2$.'
address: |
    Universidade da Beira Interior\
    Rua Marquês d’Ávila e Bolama, 6200-001 Covilhã, Portugal
author:
- 'N. Correia and R. Pacheco'
title: 'Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass-type representations'
---

Introduction {#introd}
============

The harmonicity of maps $\varphi$ from a Riemann surface $M$ into a compact Lie group $G$ with identity $e$ amounts to the flatness of one-parameter families of connections. This establishes a correspondence between such maps and certain holomorphic maps $\Phi$ into the based loop group $\Omega G$, the *extended solutions* [@Uh]. Evaluating an extended solution $\Phi$ at $\lambda=-1$ we obtain a harmonic map $\varphi$ into the Lie group. If an extended solution takes values in the group of algebraic loops $\Omega_\mathrm{alg}{G}$, the corresponding harmonic map is said to have *finite uniton number*. It is well known that all harmonic maps from the two-sphere into a compact Lie group have finite uniton number [@Uh].

Burstall and Guest [@BG] have used a method suggested by Morse theory in order to describe harmonic maps with finite uniton number from $M$ into a compact Lie group $G$ with trivial centre. One of the main ingredients in that paper is the Bruhat decomposition of the group of algebraic loops $\Omega_\mathrm{alg}{G}$. Each piece $U_\xi$ of the Bruhat decomposition corresponds to an element $\xi$ in the integer lattice $\mathfrak{I}(G)=(2\pi)^{-1} \exp^{-1}(e)\cap \mathfrak{t}$ and can be described as the unstable manifold of the energy flow on the Kähler manifold $\Omega_\mathrm{alg}{G}$. Each extended solution $\Phi:M\to \Omega_\mathrm{alg}{G}$ takes values, off some discrete subset $D$ of $M$, in one of these unstable manifolds $U_\xi$ and can be deformed, under the gradient flow of the energy, to an extended solution with values in some conjugacy class of a Lie group homomorphism $\gamma_\xi:S^1\to G$. A normalization procedure allows us to choose $\xi$ among the *canonical elements* of $\mathfrak{I}(G)$; there are precisely $2^n$ canonical elements, where $n=\mathrm{rank}(G)$, and consequently $2^n$ classes of harmonic maps. Burstall and Guest [@BG] introduced also a Weierstrass-type representation for such harmonic maps in terms of meromorphic functions on $M$. It is possible to define a similar notion of canonical element for compact Lie groups $G$ with non-trivial centre [@correia_pacheco_4; @correia_pacheco_5]. In the present paper, we will not assume any restriction on the centre of $G$.

Given an involution $\sigma$ of $G$, the compact symmetric $G$-space $N=G/G^\sigma$, where $G^\sigma$ is the subgroup of $G$ fixed by $\sigma$, can be embedded totally geodesically in $G$ via the corresponding Cartan embedding $\iota_\sigma$. Hence harmonic maps into compact symmetric spaces can be interpreted as special harmonic maps into Lie groups. For inner involutions $\sigma=\mathrm{Ad}(s_0)$, where $s_0\in G$ is the geodesic reflection at some base point $x_0\in N$, the composition of the Cartan embedding with left multiplication by $s_0$ gives a totally geodesic embedding of $G/G^\sigma$ in $G$ as a connected component of $\sqrt{e}$. Reciprocally, any connected component of $\sqrt{e}$ is a compact inner symmetric $G$-space. As shown by Burstall and Guest [@BG], any harmonic map into a connected component of $\sqrt{e}$ admits an extended solution $\Phi$ which is invariant under the involution $I(\Phi)(\lambda)=\Phi(-\lambda)\Phi(-1)^{-1}$. Off a discrete set, $\Phi$ takes values in some unstable manifold $U_\xi$ and can be deformed, under the gradient flow of the energy, to an extended solution with values in some conjugacy class of a Lie group homomorphism $\gamma_\xi:S^1\to G^\sigma$. An appropriate normalization procedure which preserves both $I$-invariance and the underlying connected component of $\sqrt{e}$ allows us to choose $\xi$ among the canonical elements of $\mathfrak{I}(G^\sigma)$. As a matter of fact, since $\sigma$ is inner, $\mathrm{rank}(G)=\mathrm{rank}(G^\sigma)$ and we have $\mathfrak{I}(G)=\mathfrak{I}(G^\sigma)$, that is the canonical elements of $\mathfrak{I}(G)$ coincide with the canonical elements of $\mathfrak{I}(G^\sigma)$. Consequently, if $G$ has trivial center, we have $2^n$ classes of harmonic maps with finite uniton number into *all* inner symmetric $G$-spaces.

The theory of Burstall and Guest [@BG] on harmonic two-spheres in compact inner symmetric $G$-spaces was extended by Eschenburg, Mare and Quast [@EMQ] to outer symmetric spaces as follows: each harmonic map from a two-sphere into an outer symmetric space $G/G^\sigma$, with outer involution $\sigma$, corresponds to an extended solution $\Phi$ which is invariant under a certain involution $T_\sigma$ induced by $\sigma$ on $\Omega G$ (see also [@GO]); $\Phi$ takes values in some unstable manifold $U_\xi$, off some discrete set; under the gradient flow of the energy any such invariant extended solution is deformed to an extended solution with values in some conjugacy class of a Lie group homomorphism $\gamma_\xi:S^1\to G^\sigma$; applying the normalization procedure of extended solutions introduced by Burstall and Guest for Lie groups, $\xi$ can be chosen among the canonical elements of $\mathfrak{I}(G^\sigma)\subsetneq \mathfrak{I}(G)$; if $G$ has trivial centre, there are precisely $2^k$ canonical homorphisms, where $k=\mathrm{rank}(G^\sigma)<\mathrm{rank}(G)$; hence there are *at most* $2^k$ classes of harmonic two-spheres in $G/G^\sigma$ if $G$ has trivial centre.

In the present paper, we will establish a more accurate classification of harmonic maps from a two-sphere into compact outer symmetric spaces. This classification takes into consideration the following crucial facts concerning extended solutions associated to harmonic maps into outer symmetric spaces: although any harmonic map from a two-sphere into an outer symmetric space $G/G^\sigma$ admits a $T_\sigma$-invariant extended solution, not all $T_\sigma$-invariant extended solutions correspond to harmonic maps into $G/G^\sigma$; on the other hand, the Burstall and Guest’s normalization procedure does not necessarily preserve $T_\sigma$-invariance.

Our strategy is the following. The existence of outer involutions of a simple Lie algebra $\mathfrak{g}$ depends on the existence of non-trivial involutions of the Dynkin diagram of $\mathfrak{g}^{\mathbb{C}}$ [@BR; @EMQ; @He; @Mu]. More precisely, if $\varrho$ is a non-trivial involution of the Dynkin diagram of $\mathfrak{g}^{\mathbb{C}}$, then it induces an outer involution $\sigma_\varrho$ of $\mathfrak{g}^{\mathbb{C}}$, which we call the *fundamental outer involution*, and, as shown by Murakami [@Mu], all the other outer involutions are, up to conjugation, of the form $\sigma_{\varrho,i}:=\mathrm{Ad}\exp\pi\zeta_i\circ \sigma_\varrho$ where each $\zeta_i$ is a certain element in the integer lattice $\mathfrak{I}(G^{\sigma_\varrho})$. Each connected component of $P^{\sigma_\varrho}=\{g\in G| \,\sigma_\varrho(g)=g^{-1}\}$ is a compact outer symmetric $G$-space associated to some involution $\sigma_\varrho$ or $\sigma_{\varrho,i}$; reciprocally, any outer symmetric space $G/G^\sigma$, with $\sigma$ equal to $\sigma_\varrho$ or $\sigma_{\varrho,i}$, can be totally geodesically embedded in the Lie group $G$ as a connected component of $P^{\sigma_\varrho}$ (see Proposition \[concomp\]). As shown in Section \[basics\], any harmonic map $\varphi$ into a connected component $N$ of $P^{\sigma_\varrho}$ admits a $T_{\sigma_\varrho}$-invariant extended solution $\Phi$; off a discrete set, $\Phi$ takes values in some unstable manifold $U_\xi$. In Section \[normprocedure\] we introduce an appropriate normalization procedure in order to obtain from $\Phi$ a *normalized* extended solution $\tilde{\Phi}$ with values in some unstable manifold $U_\zeta$ such that: $\zeta$ is a canonical element of $\mathfrak{I}(G^{\sigma_\varrho})$; $\tilde{\Phi}$ is $T_\tau$-invariant, where $\tau$ is the outer involution given by $\tau=\mathrm{Ad}\exp\pi(\xi-\zeta)\circ \sigma_{\varrho}$; $\tilde{\Phi}(-1)$ takes values in some connected component of $P^{\sigma_\varrho}$ which is an isometric copy of $N$ completely determined by $\zeta$ and $\tau$; moreover, $\tilde{\Phi}(-1)$ coincides with $\varphi$ up to isometry. Hence, we obtain a classification of harmonic maps of finite uniton number from $M$ into outer symmetric $G$-spaces in terms of the pairs $(\zeta,\tau)$.

Dorfmeister, Pedit and Wu [@DPW] have introduced a general scheme for constructing harmonic maps from a Riemann surface into a compact symmetric space from holomorphic data, in which the harmonic map equation reduces to a linear ODE similar to the classical Weierstrass representation of minimal surfaces. Burstal and Guest [@BG] made this scheme more explicit for the case $M=S^2$ by establishing a “Weierstrass formula" for harmonic maps with finite uniton number into Lie groups and their inner symmetric spaces. In Theorem \[sigmaweirstrass\] we establish a version of this formula to outer symmetric spaces, which allows us to describe the corresponding $T_\sigma$-invariant extended solutions in terms of meromorphic functions on $M$. For normalized extended solutions and “low uniton number", such descriptions are easier to obtain. In Section \[examples\] we give several explicit examples of harmonic maps from the two-sphere into classical outer symmetric spaces: Theorem \[RAs\] interprets old results by Calabi [@calabi_1967] and Eells and Wood [@eells_wood_1983] concerning harmonic spheres in real projective spaces $\mathbb{R}P^{2n-1}$ in view of our classification; harmonic two-spheres into the real Grassmannian $G_3(\mathbb{R}^6)$ are studied in detail; we show that all harmonic two spheres into the *Wu manifold* $SU(3)/SO(3)$ can be obtained explicitly by choosing two meromorphic functions on $S^2$ and then performing a finite number of algebraic operations, in agreement with the explicit constructions established by H. Ma in [@Ma].

Groups of algebraic loops
=========================

For completeness, in this section we recall some fundamental facts concerning the structure of the group of algebraic loops in a compact Lie group. Further details can be found in [@BG; @correia_pacheco_3; @PS].

Let $G$ be a compact matrix semisimple Lie group with Lie algebra $\mathfrak g$ and identity $e$. Denote the *free* and *based* loop groups of $G$ by $\Lambda G$ and $\Omega G$, respectively, whereas $\Lambda_+ G^{\mathbb{C}}$ stands for the subgroup of $\Lambda G^{\mathbb{C}}$ consisting of loops $\gamma:S^1\to G^{\mathbb{C}}$ which extend holomorphically to the unitary disc $|\lambda|<1$.

Fix a maximal torus $T$ of $G$ with Lie algebra $\mathfrak{t}\subset \mathfrak{g}$. Let $\Delta\subset \mathrm{i}\mathfrak{t}^*$ be the corresponding set of roots, where $\mathrm{i}:=\sqrt{-1}$, and, for each $\alpha\in \Delta$, denote by ${\mathfrak{g}}_\alpha$ the corresponding root space. Choose a fundamental Weyl chamber $\mathcal{W}$ in $\mathfrak{t}$, which corresponds to fix a positive root system $\Delta^+$. The intersection $\mathfrak{I}'(G):=\mathfrak{I}(G)\cap\mathcal{W}$ parameterizes the conjugacy classes of homomorphisms $S^1\to G$. More precisely, $\mathrm{Hom}(S^1,G)$ is the disjoint union of $\Omega_\xi(G)$, with $\xi\in \mathfrak{I}'(G)$, where $\Omega_\xi(G)$ is the conjugacy class of homomorphisms which contains $\gamma_\xi(\lambda)=\exp{(-\mathrm{i}\ln(\lambda)\xi)}$.

The *Bruhat decomposition* states that the subgroup of algebraic based loops $\Omega_\mathrm{alg}G$ is the disjoint union of the orbits $\Lambda^+_\mathrm{alg}G^{\mathbb{C}}\cdot\gamma_\xi$, with $\xi\in \mathfrak{I}'(G)$, where $\cdot$ denotes the dressing action of $\Lambda_+G^\mathbb{C}$ on $\Omega G$ induced by the *Iwasawa decomposition* $\Lambda G^{\mathbb{C}}\cong \Omega G\times \Lambda_+G^\mathbb{C}$. According to the Morse theoretic interpretation [@BG; @PS] of the Bruhat decomposition, for each $\xi\in\mathfrak{I}'(G)$, $U_\xi(G):=\Lambda^+_\mathrm{alg}G^{\mathbb{C}}\cdot \gamma_\xi$ is the unstable manifold of $\Omega_\xi(G)$ under the flow induced by the energy gradient vector field $-\nabla E$: each $\gamma\in U_\xi(G)$ flows to some homomorphism $u_\xi(\gamma)$ in $\Omega_\xi(G)$.

In [@BG], the authors proved that, for each $\xi\in\mathfrak{I}'(G)$, the critical manifold $\Omega_\xi(G)$ is a complex homogeneous space of $G^{\mathbb{C}}$ and the unstable manifold $U_\xi(G)$ is a complex homogeneous space of the group $\Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}$. Moreover, $U_\xi(G)$ carries a structure of holomorphic vector bundle over $\Omega_\xi(G)$ and the bundle map $u_\xi:U_\xi(G)\to \Omega_\xi(G)$ is precisely the natural projection given by $[\gamma]\mapsto [\gamma(0)]$.

Define a partial order $\preceq$ over $\mathfrak{I}'(G)$ as follows: $\xi\preceq \xi'$ if $\mathfrak{p}^{\xi}_i\subset \mathfrak{p}^{\xi'}_i$ for all $i\geq 0$, where $\mathfrak{p}_i^\xi=\sum_{j\leq i}{\mathfrak{g}}_j^\xi$ and ${\mathfrak{g}}_j^\xi$ is the $j\mathrm{i}$-eigenspace of $\mathrm{ad}{\xi}$. As shown in [@correia_pacheco_3], one can define a $\Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}$-invariant fibre bundle morphism $\mathcal{U}_{\xi,\xi'}:U_\xi(G)\to U_{\xi'}(G)$ by $$\mathcal{U}_{\xi,\xi'}(\Psi\cdot \gamma_{\xi})=\Psi\cdot\gamma_{\xi'}, \quad \Psi\in\Lambda^+_{\mathrm{alg}}G^{\mathbb{C}},$$ whenever $\xi\preceq \xi'$. Since the holomorphic structures on $U_\xi(G)$ and $U_{\xi'}(G)$ are induced by the holomorphic structure on $\Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}$, the fibre-bundle morphism $\mathcal{U}_{\xi,\xi'}$ is holomorphic.

Harmonic spheres in Lie groups
==============================

Harmonic maps from the two-sphere $S^2$ into a compact matrix Lie group $G$ can be classified in terms of certain pieces of the Bruhat decomposition of $\Omega_\mathrm{alg}G$. Next we recall briefly this theory from [@BG; @correia_pacheco_3; @correia_pacheco_4; @correia_pacheco_5].

Extended Solutions
------------------

Let $M$ be a simply-connected Riemann surface, $\varphi:M\rightarrow G$ be a smooth map and $\rho:G\to \mathrm{End}(V)$ a finite representation of $G$. Equip $G$ with a bi-invariant metric. If $\varphi$ is an harmonic map of *finite uniton number*, it admits an extended solution $\Phi:M\to \Omega G$ with $\Phi(M)\subseteq \Omega_{\mathrm{alg}}G$ and $\varphi=\Phi_{-1}$. In this case, we can write $\rho\circ\Phi=\sum_{i=r}^s\zeta_i\lambda^i$ for some $r\leq s\in\mathbb{Z}$. The number $s-r$ is called the *uniton number* of $\Phi$ with respect to $\rho$, and the minimal value of $s-r$ (with respect to all extended solutions associated to $\varphi$) is called the *uniton number* of $\varphi$ with respect to $\rho$ and it is denoted by $r_\rho(\varphi)$. As explained in [@correia_pacheco_5], this definition of uniton number of an extended solution with respect to the adjoint representation is twice that of Burstall and Guest [@BG]. K. Uhlenbeck [@Uh] proved that all harmonic maps from the two-sphere have finite uniton number. For simplicity of exposition, henceforth we will take $M=S^2$. However, all our results still hold for harmonic maps of finite uniton number from an arbitrary Riemann surface.

[@BG]\[usd\] [Let $\Phi:S^2\to \Omega_{\mathrm{alg}}G$ be an extended solution. Then there exists some $\xi\in \mathfrak{I}'(G)$, and some discrete subset $D$ of $S^2$, such that $\Phi(S^2\setminus D)\subseteq U_\xi(G)$.]{}

Given a smooth map $\Phi:S^2\setminus D\to U_\xi(G)$, consider $\Psi:S^2\setminus D \to \Lambda_{\mathrm{alg}}^+G^{\mathbb{C}}$ such that $\Phi =\Psi\cdot\gamma_\xi $, that is $\Psi\gamma_\xi=\Phi b$ for some $b:S^2\setminus D\to \Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}.$ Write $\Psi^{-1}\Psi_z=\sum_{i\geq 0} X'_i\lambda^i$, and $\Psi^{-1}\Psi_{\bar{z}}=\sum_{i\geq 0} X''_i\lambda^i.$ Proposition 4.4 in [@BG] establishes that $\Phi$ is an extended solution if, and only if, $$\label{im}
\mathrm{Im} X'_i\subset \,\mathfrak{p}^\xi_{i+1},\,\,\,\,\mathrm{Im} X''_i\subset \mathfrak{p}^\xi_{i},$$ where $\mathfrak{p}_i^\xi=\bigoplus_{j\leq i}{\mathfrak{g}}_j^\xi$ and ${\mathfrak{g}}_j^\xi$ is the $j\mathrm{i}$-eigenspace of $\mathrm{ad}{\xi}$. The derivative of the harmonic map $\varphi=\Phi_{-1}$ is given by the following formula.

\[poi\][@correia_pacheco_3] Let $\Phi=\Psi\cdot\gamma_\xi:S^2\to \Omega_{\mathrm{alg}}G$ be an extended solution and $\varphi=\Phi_{-1}:S^2\to G$ the corresponding harmonic map. Then $$\varphi^{-1}\varphi_z=-2\sum_{i\geq 0}b(0){X'_i}^{i+1}b(0)^{-1},$$ where ${X'_i}^{i+1}$ is the component of ${X'_i}$ over ${\mathfrak{g}}^\xi_{i+1}$, with respect to the decomposition ${\mathfrak{g}}^{\mathbb{C}}=\bigoplus {\mathfrak{g}}^\xi_j$.

Both the fiber bundle morphism $\mathcal{U}_{\xi,\xi'}:U_\xi(G)\to U_{\xi'}(G)$ and the bundle map $u_\xi:U_\xi(G)\to \Omega_\xi(G)$ preserve harmonicity.

[@BG; @correia_pacheco_3] \[popo\] Let $\Phi:S^2\setminus D\to U_\xi(G)$ be an extended solution. Then

1.  $u_\xi\circ\Phi:S^2\setminus D\to \Omega_\xi(G)$ is an extended solution, with $\xi\in\mathfrak{I}(G)$;

2.  for each $\xi'\in \mathfrak{I}'(G)$ such that $\xi\preceq \xi'$, $\mathcal{U}_{\xi,\xi'}(\Phi)=\mathcal{U}_{\xi,\xi'}\circ \Phi:S^2\setminus D\to U_{\xi'}(G)$ is an extended solution.

Weierstrass representation. {#weircond}
---------------------------

Taking a larger discrete subset if necessary, one obtains a more explicit description for harmonic maps of finite uniton number and their extended solutions as follows.

\[BG\][@BG] Let $\Phi:S^2\to \Omega_{\mathrm{alg}}G$ be an extended solution. There exists a discrete set $D'\supseteq D$ of $S^2$ such that $\Phi{\big|_{S^2\setminus D'}}=\exp C\cdot \gamma_\xi$ for some holomorphic vector-valued function $C: S^2\setminus D'\to \mathfrak{u}^0_\xi$, where $\mathfrak{u}^0_\xi$ is the finite dimensional nilpotent subalgebra of $\Lambda^+_{\mathrm{alg}}{\mathfrak{g}}^{\mathbb{C}}$ defined by $$\mathfrak{u}^0_\xi=\bigoplus_{0\leq i<r(\xi)}\lambda^i(\mathfrak{p}^\xi_i)^\perp,\quad (\mathfrak{p}^\xi_i)^\perp=\bigoplus _{i<j\leq r(\xi)}\mathfrak{g}_j^\xi,$$ with $r(\xi)=\mathrm{max}\{j\,|\,\,{\mathfrak{g}}_j^\xi\neq 0\}$. Moreover, $C$ can be extended meromorphically to $S^2$.

Conversely, taking account and the well-known formula for the derivative of the exponential map, we see that if $C: S^2\to \mathfrak{u}^0_\xi$ is meromorphic then $\Phi=\exp{C}\cdot \gamma_{\xi}$ is an extended solution if and only if in the expression $$\label{C_z}
(\exp C)^{-1}(\exp C)_z=C_z-\frac{1}{2!}(\mathrm{ad} C)C_z+\ldots +(-1)^{r(\xi)-1}\frac{1}{r(\xi)!}(\mathrm{ad} C)^{r(\xi)-1}C_z,$$ the coefficient $\lambda^i$ have zero component in each ${\mathfrak{g}}_{i+2}^\xi,\ldots,{\mathfrak{g}}^\xi_{r(\xi)}$.

$S^1$-invariant extended solutions
----------------------------------

Extended solutions with values in some $\Omega_\xi(G)$, off a discrete subset, are said to be *$S^1$-invariant*. If we take a unitary representation $\rho:G \to U(n)$ for some $n$, then for any such extended solution $\Phi$ we have $\rho\circ \Phi_\lambda=\sum_{i=r}^s\lambda^i\pi_{W_i},$ where, for each $i$, $\pi_{W_i}$ is the orthogonal projection onto a complex vector subbundle $W_i$ of $\underline{{\mathbb{C}}}^n:=M\times {\mathbb{C}}^n$ and $\underline{{\mathbb{C}}}^n=\bigoplus_{i=r}^sW_i$ is an orthogonal direct sum decomposition. Set $A_i=\bigoplus_{j\leq i }W_j$ so that $$\{0\}\subset A_r \subset \ldots \subset A_{i-1}\subset A_i\subset A_{i+1}\subset\ldots \subset A_s= \underline{{\mathbb{C}}}^n.$$ The harmonicity condition amounts to the following conditions on this flag: for each $i$, $A_i$ is a holomorphic subbundle of $\underline{{\mathbb{C}}}^n$; the flag is *superhorizontal*, in the sense that, for each $i$, we have $\partial A_i\subseteq A_{i+1}$, that is, given any section $s$ of $A_i$ then $\frac{\partial s}{\partial z}$ is a section of $A_{i+1}$ for any local complex coordinate $z$ of $S^2$.

Normalization of harmonic maps {#BGnorm}
------------------------------

Let $\Delta_0:=\{\alpha_1,\ldots,\alpha_r\}\subset \Delta^+$ be the basis of positive simple roots, with dual basis $\{H_1,\ldots, H_r\}\subset\mathfrak{t}$, that is $\alpha_i(H_j)=\mathrm{i}\,\delta_{ij}$, where $r=\mathrm{rank}(\mathfrak{g})$. Given $\xi=\sum n_iH_i$ and $\xi'=\sum n'_iH_i$ in $\mathfrak{I}'({G})$, we have $n_i,n'_i\geq 0$ and observe that $\xi\preceq\xi'$ if and only if $n'_i\leq n_i$ for all $i$. For each $I\subseteq \{1,\ldots,r\}$, define the cone $\mathfrak{C}_{I}=\Big\{\sum_{i=1}^r n_i H_i|\, n_i\geq 0, \,\mbox{$n_j=0$ iff $j\notin I$}\Big\}.$

[@correia_pacheco_5] Let $\xi\in\mathfrak{I}'({G})\cap \mathfrak{C}_{I}$. We say that $\xi$ is a *$I$-canonical element* of $G$ with respect to $\mathcal{W}$ if it is a maximal element of $(\mathfrak{I}'({G})\cap \mathfrak{C}_{I},\preceq)$, that is: if $\xi\preceq \xi'$ and $\xi'\in \mathfrak{I}'({G})\cap \mathfrak{C}_{I}$ then $\xi=\xi'$.

When $G$ has trivial centre, which is the case considered in [@BG], there exists a unique $I$-canonical element, which is given by $\xi_I=\sum_{i\in I}H_i$, for each $I$. When $G$ has non-trivial centre, the $I$-canonical elements of $G$ were described in [@correia_pacheco_4; @correia_pacheco_5].

Any harmonic map $\varphi:S^2\to G$ admits a *normalized extended solution*, that is, an extended solution $\Phi$ taking values in $U_\xi(G)$, off some discrete set, for some canonical element $\xi$. This is a consequence of the following generalization of Theorem 4.5 in [@BG].

\[nor\][@correia_pacheco_3] Let $\Phi:S^2\setminus D\to U_\xi(G)$ be an extended solution. Take $\xi'\in \mathfrak{I}'({G})$ such that $\xi\preceq {\xi'}$ and ${\mathfrak{g}}_0^\xi={\mathfrak{g}}_0^{\xi'}$. Then $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is a constant loop in $\Omega_{\mathrm{alg}}{G}$ and $\gamma\Phi:S^2\setminus D\to U_{\xi'}(G)$.

The uniton number of a normalized extended solution $\Phi:S^2\setminus D\to U_\xi(G)$ can be computed with respect to any irreducible $n$-dimensional representation ${\rho}:G\to\mathrm{End}(V)$ with highest weight $\omega^*$ and lowest weight $\varpi^*$ as follows [@correia_pacheco_5]: $r_\rho(\xi):=\omega^*(\xi)-\varpi^*(\xi)$.

Harmonic spheres in outer symmetric spaces
==========================================

In the following sections we will establish our classification of harmonic maps from $S^2$ into compact outer symmetric spaces and establish a Weierstrass formula for such harmonic maps. These will allow us to produce some explicit examples of harmonic maps from two-spheres into outer symmetric spaces from meromorphic data.

As we have referred in Section \[introd\], although any harmonic map from a two-sphere into an outer symmetric space $G/K$ admits a $T_\sigma$-invariant extended solution, not all $T_\sigma$-invariant extended solutions correspond to harmonic maps into $G/K$; by Proposition \[concomp\] and Theorem \[GO\] below, they correspond to a harmonic map into some possibly different outer symmetric space $G/K'$ (compare Theorem \[RAs\] with Theorem \[36\] for an example where this happens).

Symmetric $G$-spaces and Cartan embeddings
------------------------------------------

Let $N=G/K$ be a symmetric space, where $K$ is the isotropy subgroup at the base point $x_0\in N$, and let $\sigma:G\to G$ be the corresponding involution: we have $G^{\sigma}_0\subseteq K\subseteq G^{\sigma},$ where $G^{\sigma}$ is the subgroup fixed by $\sigma$ and $G_0^{\sigma}$ denotes its connected component of the identity. We assume that $N$ is a *bottom space*, i.e. $K=G^{\sigma}$. Let $\mathfrak{g}=\mathfrak{k}_\sigma\oplus \mathfrak{m}_\sigma$ be the $\pm1$-eigenspace decomposition associated to the involution $\sigma$, where $\mathfrak{k}_\sigma$ is the Lie algebra of $K$. Consider the (totally geodesic) *Cartan embedding* $\iota_\sigma:N\hookrightarrow G$ defined by $\iota_\sigma (g\cdot x_0)=g\sigma(g^{-1})$. The image of the Cartan embedding is precisely the connected component $P^\sigma_e$ of $P^{\sigma}:=\{g\in G|\,\sigma(g)=g^{-1}\}$ containing the identity $e$ of the group $G$. Observe that, given $\xi\in \mathfrak{I}(G)\cap \mathfrak{k}_\sigma$, then $\exp(\pi\xi)\in P^\sigma$. We denote by $P_{\xi}^{\sigma}$ the connected component of $P^{\sigma}$ containing $\exp(\pi\xi)$.

\[concomp\] Given $\xi\in \mathfrak{I}(G)\cap \mathfrak{k}_\sigma$, we have the following.

1.  $G$ acts transitively on $P_{\xi}^\sigma$ as follows: for $g\in G$ and $h\in  P_{\xi}^{\sigma}$, $$\label{gaction}
    g\cdot_\sigma h=gh\sigma(g^{-1}).$$

2.  $P^{\sigma}_{\xi}$ is a bottom symmetric $G$-space totally geodesically embedded in $G$ with involution $$\label{tau}
    \tau=\mathrm{Ad}(\exp\pi\xi)\circ \sigma.$$

3.  For any other $\xi'\in \mathfrak{I}(G)\cap\mathfrak{k}_\sigma$ we have $\exp(\pi\xi')\in P^{\tau}$ and $P^\tau_{\xi'}=\exp(\pi\xi) P_{\xi'-\xi}^\sigma.$

4.  The $\pm 1$-eigenspace decomposition $\mathfrak{g}=\mathfrak{k}_\tau\oplus\mathfrak{m}_\tau$ associated to the symmetric $G$-space $P_{\xi}^\sigma$ at the fixed point $\exp(\pi\xi)\in P_{\xi}^\sigma$ is given by $$\begin{aligned}
    \label{hc} \mathfrak{k}_\tau^{\mathbb{C}}&= \bigoplus \mathfrak{g}^\xi_{2i}\cap \mathfrak{k}_\sigma^{\mathbb{C}}\oplus \bigoplus\mathfrak{g}^\xi_{2i+1}\cap \mathfrak{m}_\sigma^{\mathbb{C}}\\  \label{pc} \mathfrak{m}_\tau^{\mathbb{C}}&=\bigoplus \mathfrak{g}^\xi_{2i+1}\cap \mathfrak{k}_\sigma^{\mathbb{C}}\oplus \bigoplus\mathfrak{g}^\xi_{2i}\cap \mathfrak{m}_\sigma^{\mathbb{C}}.\end{aligned}$$

Take $h\in P^\sigma$. We have $$\sigma(g\cdot_\sigma h)=\sigma(gh\sigma(g^{-1}))=\sigma(g)h^{-1}g^{-1}=(gh\sigma(g^{-1}))^{-1}=(g\cdot_\sigma h)^{-1}.$$ Then $g\cdot_\sigma h\in P^\sigma$ and we have a continuous action of $G$ on $P^\sigma$. Since $G$ is connected, this action induces an action of $G$ on each connected component of $P^\sigma$. Since $g\cdot_\sigma e=g\sigma(g^{-1})=\iota_\sigma(g\cdot x_0)$ and $\iota_\sigma(N)=P_e^\sigma$, the action $\cdot_\sigma$ of $G$ on $P_e^\sigma$ is transitive.

Take $\xi\in \mathfrak{I}(G)\cap \mathfrak{k}_\sigma$, so that $\sigma(\xi)=\xi$ and $\exp2\pi\xi=e$. Consider the involution $\tau$ defined by . If $g\in P^\sigma$, then $$\tau(\exp(\pi\xi)g)=\exp(\pi\xi)\sigma(\exp(\pi\xi)g)\exp(\pi\xi)=\sigma(g)\exp(\pi\xi)=(\exp(\pi\xi)g)^{-1},$$ which means that $\exp(\pi\xi)g\in P^\tau$. Reciprocally, if $\exp(\pi\xi)g\in P^\tau$, one can check similarly that $g\in P^{\sigma}$. Hence $P^\tau=\exp(\pi\xi)P^\sigma$. In particular, by continuity, $P^\tau_{\xi'}=\exp(\pi\xi) P^\sigma_{\xi'-\xi}$ for any other $\xi'\in \mathfrak{I}(G)$ with $\sigma(\xi')=\xi'$.

Reversing the rules of $\sigma= \mathrm{Ad}(\exp\pi\xi)\circ \tau$ and $\tau$, we also have $P^\sigma_{\xi}=\exp(\pi\xi)P_e^\tau.$ Since $G$ acts transitively on $P_e^\tau$, for each $h\in P^\sigma_{\xi}$ there exists $g\in G$ such that $$h=\exp(\pi\xi) (g\cdot_\tau e)=(\exp(\pi\xi)g)\cdot_\sigma \exp(\pi\xi).$$ This shows that $G$ also acts transitively on $P^\sigma_{\xi}$. The isotropy subgroup at $\exp(\pi\xi)$ consists of those elements $g$ of $G$ satisfying $g\exp(\pi\xi)\sigma(g^{-1})=\exp(\pi\xi)$, that is those elements $g$ of $G$ which are fixed by $\tau$: $$\label{isot}
\exp(\pi\xi)\sigma(g)\exp(\pi\xi)=g.$$ Hence $P^\sigma_{\xi}\cong G/G^{\tau},$ which is a bottom symmetric $G$-space with involution $\tau$. Since $P_e^\tau\subset G$ totally geodesically and $P^\sigma_\xi$ is the image of $P_e^\tau$ under an isometry (left multiplication by $\exp\pi\xi$), then $P_{\xi}^\sigma\subset G$ totally geodesically.

Differentiating at the identity we get $\mathfrak{k}_\tau=\{X\in \mathfrak{g}|\, X=\mathrm{Ad}(\exp\pi\xi)\circ\sigma(X)\}.$ Taking account of the formula $\mathrm{Ad}({\exp (\pi\xi)})=e^{\pi \mathrm{ad} \xi}$ and that $\sigma$ commutes with $\mathrm{ad}\xi$, we obtain ; and follows similarly.

### Outer symmetric spaces.

The existence of outer involutions of a simple Lie algebra $\mathfrak{g}$ depends on the existence of non-trivial involutions of the Dynkin diagram of $\mathfrak{g}^{\mathbb{C}}$ [@BR; @EMQ; @He; @Mu]. Fix a maximal abelian subalgebra $\mathfrak{t}$ of $\mathfrak{g}$ and a Weyl chamber $\mathcal{W}$ in $\mathfrak{t}$, which amounts to fix a system of positive simple roots $\Delta_0=\{\alpha_1,\ldots,\alpha_r\}$, where $r=\mathrm{rank}(\mathfrak{g})$. Let $\varrho$ be a non-trivial involution of the Dynkin diagram and $\sigma_\varrho$ the *fundamental outer involution* associated to $\varrho$ [@BR; @Mu]. The (local isometry classes of) outer symmetric spaces of compact type associated to involutions of the form $\sigma_\varrho$ are precisely

> $SU(2n)/Sp(n)$, $SU(2n+1)/SO(2n+1)$, $E_6/F_4$ and the real projective spaces $\mathbb{R}P^{2n-1}$.

These spaces are called the *fundamental outer symmetric spaces*. The remaining classes of outer involutions are obtained as follows [@BG; @Mu]. Let $\mathfrak{g}=\mathfrak{k}_{\varrho}\oplus \mathfrak{m}_{\varrho}$ be the corresponding $\pm 1$-eigenspace decomposition of $\mathfrak{g}$. As shown in Proposition 3.20 of [@BR], the Lie subalgebra $\mathfrak{k}_{\varrho}$ is simple and the orthogonal projection of $\Delta_0$ onto $\mathfrak{k}_{\varrho}$, $\pi_{\mathfrak{k}_{\varrho}}(\Delta_0)$, is a basis of positive simple roots of $\mathfrak{k}_\varrho$ associated to the maximal abelian subalgebra $\mathfrak{t}_{\mathfrak{k}_{\varrho}}:=\mathfrak{t}\cap \mathfrak{k}_{\varrho}$. Consider the split $\mathfrak{t}=\mathfrak{t}_{{\mathfrak{k}_\varrho}}\oplus \mathfrak{t}_{\mathfrak{m}_\varrho}$ with respect to $\mathfrak{g}={\mathfrak{k}_\varrho}\oplus {\mathfrak{m}_\varrho}$. Set $s=r-k$, where $k=\mathrm{rank}(\mathfrak{k}_\varrho)$. We can label the basis $\Delta_0$ in order to get the following relations: $\varrho (\alpha_j)=\alpha_j$ for $1\leq j\leq k-s$ and $\varrho (\alpha_j)=\alpha_{s+j}$ for $k-s+1\leq j\leq k$. Let $\pi_{\mathfrak{k}_\varrho}$ be the orthogonal projection of $\mathfrak{t}$ onto $\mathfrak{t}_{{\mathfrak{k}_\varrho}}$, that is $\pi_{\mathfrak{k}_\varrho}(H)=\frac12(H+\sigma_\varrho (H))$ for all $H\in \mathfrak{t}$. Set $\pi_{{\mathfrak{k}_\varrho}}(\Delta_0)=\{\beta_1,\ldots,\beta_k\}$, with $$\label{bes}
\beta_j=\left\{\begin{array}{cl} \alpha_j & \mbox{for $1\leq j\leq k-s$}
            \\ \frac12(\alpha_j+\alpha_{j+s}) & \mbox{for $k-s+1\leq j\leq k$}
\end{array}\right..$$ This is a basis of $\mathrm{i}\mathfrak{t}_{\mathfrak{k}_\varrho}^*$ with dual basis $\{\zeta_1,\ldots, \zeta_k\}$ given by $$\label{zes}
\zeta_j=\left\{\begin{array}{cl}
            H_j & \mbox{for $1\leq j\leq k-s$}  \\
           H_j+H_{j+s} & \mbox{for $k-s+1\leq j\leq k$}
          \end{array}\right..$$

[@Mu]\[murak\] Let $\varrho$ be an involution of the Dynkin diagram of $\mathfrak{g}$. Let $$\omega=\sum_{j=1}^{k-s}n_j\beta_j+\sum_{j=k-s+1}^{k}n'_j\beta_j$$ be the highest root of $\mathfrak{k}_\varrho$ with respect to $\pi_{\mathfrak{k}_{\varrho}}(\Delta_0)=\{\beta_1,\ldots,\beta_k\}$, defined as in . Given $i$ such that $n_i=1$ or $2$, define an involution $\sigma_{\varrho,i}$ by $$\label{sigmasis}
  \sigma_{\varrho,i}=\mathrm{Ad}(\exp\pi \zeta_i)\circ \sigma_\varrho.$$ Then any outer involution of $\mathfrak{g}$ is conjugate in $\mathfrak{Aut}(\mathfrak{g})$, the group of automorphisms of $\mathfrak{g}$, to some $\sigma_\varrho$ or $\sigma_{\varrho,i}$. In particular, there are at most $k-s+1$ conjugacy classes of outer involutions.

The list of all (local isometry classes of) irreducible outer symmetric spaces of compact type is shown in Table 1 (cf. [@BR; @EMQ; @He]).

                                                 $\mathrm{rank}(G)$   $\mathrm{rank}(K)$   $\mathrm{rank}(G/K)$   $\mathrm{dim}(G/K)$
  --------------------------------------------- -------------------- -------------------- ---------------------- ---------------------
                 $SU(2n)/SO(2n)$                       $2n-1$                $n$                  $2n-1$             $(2n-1)(n+1)$
               $SU(2n+1)/SO(2n+1)$                      $2n$                 $n$                   $2n$                $n(2n+3)$
                 $SU(2n)/Sp(n)$                        $2n-1$                $n$                  $ n-1$             $(n-1)(2n+1)$
   $G_{p}(\mathbb{R}^{2n})$ ($p$ odd $\leq n$)          $n$                 $n-1$                  $p$                 $p(2n-p)$
                   $E_6/Sp(4)$                          $6$                  $4$                   $6$                   $42$
                    $E_6/F_4$                           $6$                  $4$                   $2$                   $26$

  : Irreducible outer symmetric spaces.

Given an outer involution $\sigma$ of the form $\sigma_{\varrho,i}$ or $\sigma_\varrho$ and its $\pm 1$-eigenspace decomposition ${\mathfrak{g}}=\mathfrak{k}_\sigma\oplus \mathfrak{m}_\sigma$, set $\mathfrak{t}_{\mathfrak{k}_\sigma}=\mathfrak{t}\cap \mathfrak{k}_\sigma$, which is a maximal abelian subalgebra of $\mathfrak{k}_\sigma$. Following [@EMQ], a non-empty intersection of $\mathfrak{t}_{\mathfrak{k}_\sigma}$ with a Weyl chamber in $\mathfrak{t}$ is called a *compartment*. Each compartment lies in a Weyl chamber in $\mathfrak{t}_{\mathfrak{k}_\sigma}$ and the Weyl chambers in $\mathfrak{t}_{\mathfrak{k}_\sigma}$ can be decomposed into the same number of compartments [@EMQ].

The intersection of the integer lattice $\mathfrak{I}(G)$ with the Weyl chamber $\mathcal{W}$ in $\mathfrak{t}$, which we have denoted by $\mathfrak{I}'(G)$, is described in terms of the dual basis $\{H_1,\ldots,H_r\}\subset \mathfrak{t}$, with $r=\mathrm{rank} (\mathfrak{g})$, by $$\mathfrak{I}'(G)=\big\{\sum_{i=1}^r n_iH_i\in \mathfrak{I}(G) |\,\mbox{$n_i\in \mathbb{N}_0$ for all $i$}\big\}.$$ When $\sigma$ is a fundamental outer involution $\sigma_\varrho$, the compartment $\mathcal{W}\cap \mathfrak{t}_{\mathfrak{k_\varrho}}$ is itself a Weyl chamber in $\mathfrak{t}_{\mathfrak{k_\varrho}}$. Then, the intersection of the integer lattice $\mathfrak{I}(G^{\sigma_\varrho})$ with the Weyl chamber $\mathcal{W}\cap \mathfrak{t}_{\mathfrak{k_\varrho}}$, is given by $$\mathfrak{I}'(G^{\sigma_\varrho})=\big\{\sum_{i=1}^k n_i\zeta_i\in \mathfrak{I}(G) |\,\mbox{$n_i\in \mathbb{N}_0$ for all $i$}\big\}=\mathfrak{I}'(G)\cap \mathfrak{t}_{\mathfrak{k_\varrho}}.$$

### Cartan embeddings of fundamental outer symmetric spaces.

Next we describe those elements $\xi$ of $ \mathfrak{I}'(G^{\sigma_\varrho})$ for which the connected component $P_\xi^{\sigma_\varrho}$ of $P^{\sigma_\varrho}$ containing $\exp(\pi\xi)$ can be identified with the fundamental outer symmetric $G$-space associated to $\varrho$. Start by considering the following $\sigma_\varrho$-invariant subsets of the root system $\Delta\subset \mathrm{i}\mathfrak{t}^*$ of $\mathfrak{g}$: $$\begin{aligned}
\label{deltas}
 \Delta(\mathfrak{k}_\varrho)=\{\alpha\in\Delta|\, {\mathfrak{g}}_\alpha\subset \mathfrak{k}_\varrho^{\mathbb{C}}\},\,\,
   \Delta(\mathfrak{m}_\varrho)=\{\alpha\in\Delta|\, {\mathfrak{g}}_\alpha\subset \mathfrak{m}_\varrho^{\mathbb{C}}\},\,\, \Delta_\varrho=\Delta\setminus  \left(\Delta(\mathfrak{k}_\varrho)\cup  \Delta(\mathfrak{m}_\varrho)\right).\end{aligned}$$ Then $$\mathfrak{k}_\varrho^{\mathbb{C}}=\mathfrak{t}^{\mathbb{C}}_{\mathfrak{k}_\varrho}\oplus\pi_{\mathfrak{k}_\varrho}(\mathfrak{r}_\varrho)\oplus\bigoplus_{\alpha\in \Delta(\mathfrak{k}_\varrho)}{\mathfrak{g}}_\alpha,\,\,\,  \mathfrak{m}_\varrho^{\mathbb{C}}=\mathfrak{t}^{\mathbb{C}}_{\mathfrak{m}_\varrho}\oplus\pi_{\mathfrak{m}_\varrho}(\mathfrak{r}_\varrho)\oplus\bigoplus_{\alpha\in \Delta(\mathfrak{m}_\varrho)}{\mathfrak{g}}_\alpha,$$ where $\mathfrak{r}_\varrho=\bigoplus_{\alpha\in\Delta_\varrho}{\mathfrak{g}}_\alpha$. Since the involution $\varrho$ acts on $\Delta_\varrho$ as a permutation without fixed points, we can fix some subset $\Delta'_\varrho$ so that $\Delta_\varrho$ is the disjoint union of $\Delta'_\varrho$ with $\varrho(\Delta'_\varrho)$: $$\label{delta1}
  \Delta_\varrho= \Delta'_\varrho {\sqcup}\,\, \varrho(\Delta'_\varrho).$$ For each $\alpha \in\Delta'_\varrho$, $\sigma_\varrho$ restricts to an involution in the subspace ${\mathfrak{g}}_\alpha\oplus {\mathfrak{g}}_{\varrho(\alpha)}\subset \mathfrak{r}_\varrho$. Hence we have the following.

\[pmmm\]The orthogonal projections of $\mathfrak{r}_\varrho$ onto $\mathfrak{k}_\varrho^{\mathbb{C}}$ and $\mathfrak{m}_\varrho^{\mathbb{C}}$ are given by $$\pi_{\mathfrak{k}_\varrho}(\mathfrak{r}_\varrho)=\!\bigoplus_{\alpha\in\Delta'_\varrho}\!  \mathfrak{k}_\varrho^{\mathbb{C}}\cap \big({\mathfrak{g}}_\alpha\oplus{\mathfrak{g}}_{\varrho(\alpha)}\big),\,\,\pi_{\mathfrak{m}_\varrho}(\mathfrak{r}_\varrho)=\!\!\bigoplus_{\alpha\in\Delta'_\varrho}\!  \mathfrak{m}_\varrho^{\mathbb{C}}\cap \big({\mathfrak{g}}_\alpha\oplus{\mathfrak{g}}_{\varrho(\alpha)}\big),$$ and, for each $\alpha\in\Delta'_\varrho$, $$\mathfrak{k}_\varrho^{\mathbb{C}}\cap \big({\mathfrak{g}}_\alpha\oplus{\mathfrak{g}}_{\sigma_\varrho(\alpha)}\big)=\{X_\alpha+\sigma_\varrho(X_\alpha)|\, X_\alpha\in {\mathfrak{g}}_\alpha\},\,\,\, \mathfrak{m}_\varrho^{\mathbb{C}}\cap \big({\mathfrak{g}}_\alpha\oplus{\mathfrak{g}}_{\sigma(\alpha)}\big)=\{X_\alpha-\sigma_\varrho(X_\alpha)|\, X_\alpha\in {\mathfrak{g}}_\alpha\}.$$ In particular, $\dim \mathfrak{r}_\varrho=2\dim \pi_{\mathfrak{k}_\varrho}(\mathfrak{r}_\varrho)= 2\dim  \pi_{\mathfrak{m}_\varrho}(\mathfrak{r}_\varrho)$.

\[fundcan\] Consider the dual basis $\{\zeta_1,\ldots,\zeta_k\}$ defined by . Given $\xi \in \mathfrak{I}'(G^{\sigma_\varrho})$ with $\xi=\sum_{i=1}^kn_i\zeta_i$ and $n_i\geq 0$, then $P_\xi^{\sigma_\varrho}$ is a fundamental outer symmetric space with involution (conjugated to) $\sigma_\varrho$ if and only if $n_i$ is even for each $1\leq i\leq k-s$.

There is only one class of outer symmetric $SU(2n+1)$-spaces and, in this case, the involution $\varrho$ does not fix any simple root, that is $k-s=0$. Hence the result trivially holds for $N=SU(2n+1)/SO(2n+1).$

Next we consider the remaining fundamental outer symmetric spaces, which are precisely the symmetric spaces of *rank-split type* [@EMQ], those satisfying $\Delta(\mathfrak{m}_\varrho)=\emptyset$. For such symmetric spaces, the reductive symmetric term $\mathfrak{m}_\varrho$ satisfies $\mathfrak{m}_\varrho=\mathfrak{t}_{\mathfrak{m}_\varrho}\oplus\pi_{\mathfrak{m}_\varrho}(\mathfrak{r}_\varrho)$. On the other hand, in view of , we have, for $\tau=\mathrm{Ad}(\exp\pi\xi)\circ\sigma_\varrho$, $$\begin{aligned}
\nonumber
    \mathfrak{m}_\tau^{\mathbb{C}}&=\bigoplus \mathfrak{g}^\xi_{2i+1}\cap \mathfrak{k}^{\mathbb{C}}_\varrho\oplus \bigoplus\mathfrak{g}^\xi_{2i}\cap \mathfrak{m}^{\mathbb{C}}_\varrho\\&=
    \mathfrak{t}^{\mathbb{C}}_{\mathfrak{m}_\varrho}\oplus\!\!\! \bigoplus_{{\alpha\in \Delta(\mathfrak{k}_\varrho)\cap \Delta_\xi^-}}\!\!\! {\mathfrak{g}}_\alpha\oplus
   \!\!\!  \bigoplus_{{\alpha\in \Delta'_\varrho\cap \Delta_\xi^-}} \!\!\!\mathfrak{k}^{\mathbb{C}}_\varrho\cap({\mathfrak{g}}_\alpha\oplus  {\mathfrak{g}}_{\varrho(\alpha)})\oplus \!\!\!\bigoplus_{{\alpha\in \Delta'_\varrho\cap\Delta_\xi^+}} \!\!\!\mathfrak{m}^{\mathbb{C}}_\varrho\cap({\mathfrak{g}}_\alpha\oplus  {\mathfrak{g}}_{\varrho(\alpha)}),
  \end{aligned}$$ where $\Delta_\xi^+:=\{\alpha\in\Delta|\,\mbox{$\alpha(\xi)\mathrm{i}$ is even}\}$ and $\Delta_\xi^-:=\{\alpha\in\Delta|\,\mbox{$\alpha(\xi)\mathrm{i}$ is odd}\}$. Taking into account Lemma \[pmmm\], from this we see that $\dim \mathfrak{m}_\tau=\dim \mathfrak{m}_\varrho$ (which means, by Table 1, that $P_\xi^{\sigma_\varrho}$ is a fundamental outer symmetric space with involution conjugated to $\sigma_\varrho$) if and only if $$\bigoplus_{{\alpha\in \Delta(\mathfrak{k}_\varrho)\cap \Delta_\xi^-}}\!\! {\mathfrak{g}}_\alpha=\{0\},$$ which holds if and only if $\xi=\sum_{i=1}^kn_i\zeta_i$ with $n_i$ even for each $1\leq i\leq k-s$.

Harmonic spheres in symmetric $G$-spaces {#basics}
----------------------------------------

Given an involution $\sigma$ on $G$, define an involution $T_\sigma$ on $\Omega G$ by $T_\sigma(\gamma)(\lambda)=\sigma(\gamma(-\lambda)\gamma(-1)^{-1}).$ Let $\Omega^\sigma G$ be the fixed set of $T_\sigma$.

\[GO\][@EMQ; @GO] Given $\xi\in \mathfrak{I}(G)\cap\mathfrak{k}_\sigma$, any harmonic map $\varphi:S^2\to P^{\sigma}_\xi\subset G$ admits an $T_\sigma$-invariant extended solution $\Phi:S^2\to\Omega^\sigma G$. Conversely, given an $T_\sigma$-invariant extended solution $\Phi$, the smooth map $\varphi=\Phi_{-1}$ from $S^2$ is harmonic and takes values in some connected component of $P^{\sigma}$.

[@EMQ] \[incariance\] Given $\Phi\in U_\xi^\sigma(G):=U_\xi(G)\cap \Omega^{\sigma}G$, with $\xi\in \mathfrak{I}(G)\cap \mathfrak{k}_\sigma$, set $\gamma=u_\xi\circ \Phi$. Then $\gamma$ takes values in $K$. By continuity, $\Phi_{-1}$ and $\gamma(-1)$ take values in the same connected component of $P^\sigma$.

Hence, together with Theorems \[usd\] and \[GO\], this implies the following.

\[tinva\] Any harmonic map $\varphi$ from $S^2$ into a connected component of $P^{\sigma}$ admits an extended solution $\Phi:S^2\setminus D\to U_\xi^{\sigma}(G):=U_\xi(G)\cap \Omega^{\sigma}G$, for some $\xi\in \mathfrak{I}'(G)\cap\mathfrak{k}_\sigma$ and some discrete subset $D$. If $\sigma=\sigma_\varrho$ is the fundamental outer involution, then $\varphi=\Phi_{-1}$ takes values in $P_\xi^{\sigma_\varrho}$.

By Proposition \[incariance\], $\Phi$ and $\gamma:=u_\xi\circ \Phi$ take values in the same connected component of $P^{\sigma}$ when evaluated at $\lambda=-1$. Since $\gamma:S^1\to G^\sigma$ is a homomorphism, $\gamma$ is in the $G^\sigma$-conjugacy class of $\gamma_{\xi'}$ for some $\xi'\in \mathfrak{I}'(G^\sigma)$, where $G^\sigma$ is the subgroup of $G$ fixed by $\sigma$. Consequently, $\gamma(-1)=g\gamma_{\xi'}(-1)g^{-1}=g\cdot_\sigma \gamma_{\xi'}(-1),$ for some $g\in G^{\sigma}$, which means that $\gamma(-1)$ takes values in the connected component $P_{\xi'}^{\sigma}$. On the other hand, $\gamma$ is in the $G$-conjugacy class of $\gamma_\xi$, with $\xi\in \mathfrak{I}'(G)\cap\mathfrak{k}_\sigma$. If $\sigma$ is the fundamental outer involution $\sigma_\varrho$, then $\mathfrak{I}'(G^{\sigma})=\mathfrak{I}'(G)\cap\mathfrak{k}_\sigma$; and we must have $\xi=\xi'$.

If $\sigma$ is not a fundamental outer involution, each Weyl chamber $\mathcal{W}_\sigma$ in $\mathfrak{t}_{\mathfrak{k}_\sigma}$ can be decomposed into more than one compartment: $ \mathcal{W}_\sigma=C_1\sqcup \ldots \sqcup C_l$, where $C_1=\mathcal{W}\cap \mathfrak{t}_{\mathfrak{k}_\sigma}$ and the remaining compartments are conjugate to $C_1$ under $G$ [@EMQ], that is, there exists $g_i\in G$ satisfying $C_i=\mathrm{Ad}(g_i)(C_1)$ for each $i$. Hence, if we have an extended solution $\Phi:S^2\setminus D\to U_\xi^{\sigma}(G)$ with $\xi\in \mathfrak{I}'(G)\cap\mathfrak{k}_\sigma\subset C_1$, the corresponding harmonic map $\Phi_{-1}$ takes values in one of the connected components $P_{g_i\xi g_i^{-1}}^\sigma$.

### $\varrho$-canonical elements.

Let $I$ be a subset of $\{1,\ldots,k\}$, with $k=\mathrm{rank}(\mathfrak{k}_\varrho)$, and set $$\mathfrak{C}^\varrho_{I}=\Big\{\sum_{i=1}^k n_i \zeta_i|\, n_i\geq 0, \,\mbox{$n_j=0$ iff $j\notin I$}\Big\}.$$ Let $\xi\in\mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_{I}$. We say that $\zeta$ is a *$\varrho$-canonical element* of $G$ (with respect to the choice of $\mathcal{W}$) if $\zeta$ is a maximal element of $(\mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_{I},\preceq)$, that is: if $\zeta\preceq \zeta'$ and $\zeta'\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_{I}$ then $\zeta=\zeta'$.

When $G$ has trivial centre, the duals $\zeta_1,\ldots,\zeta_k$ belong to the integer lattice. Then, for each $I$ there exists a unique $\varrho$-canonical element, which is given by $\zeta_I=\sum_{i\in I}\zeta_i$. In this case, our definition of $\varrho$-canonical element coincides with that of $S$-canonical element in [@EMQ].

Now, consider a fundamental outer involution $\sigma_\varrho$ and let $N$ be an associated outer symmetric $G$-space, that is, $N$ corresponds to an involution of $G$ of the form $\sigma_\varrho$ or $\sigma_{\varrho,i}$, with $\zeta_i$ in the conditions of Theorem \[murak\]. If $G$ has trivial centre, we certainly have $\zeta_i\in \mathfrak{I}'({G}^{\sigma_\varrho})$. As a matter of fact, as we will see later, in most cases we have $\zeta_i\in \mathfrak{I}'({G}^{\sigma_\varrho})$, whether $G$ has trivial centre or not, with essentially one exception: for $G=SU(2n)$ and $N=SU(2n)/SO(2n)$. So, we will treat this case separately and assume henceforth that $\zeta_i\in \mathfrak{I}'({G}^{\sigma_\varrho})$.

Consider the Dynkin diagram of $\mathfrak{e}_6$:

(-0.55,-0.5)(4.82,1.30) (0,0)(1,0) (1,0)(2,0) (2,0)(3,0) (3,0)(4,0) (2,0)(2,1) (-0.1,-0.16)[$\alpha_1$]{} (1.5,1.14)[$\alpha_2$]{} (0.92,-0.16)[$\alpha_3$]{} (1.93,-0.16)[$\alpha_4$]{} (2.93,-0.16)[$ \alpha_5 $]{} (3.97,-0.16)[$\alpha_6$]{}

(0,0) (1,0) (2,0) (3,0) (4,0) (2,1)

This admits a unique nontrivial involution $\varrho$. Let $\{H_1,\ldots,H_6\}$ be the dual basis of $\Delta_0=\{\alpha_1,\ldots,\alpha_6\}$. The semi-fundamental basis $\pi_{\mathfrak{k}_\varrho}(\Delta_0)=\{\beta_1,\beta_2,\beta_3,\beta_4\}$ is given by $\beta_1=\alpha_2$, $\beta_2=\alpha_4$, $\beta_3=\frac{\alpha_1+\alpha_6}{2}$ and $\beta_4=\frac{\alpha_3+\alpha_5}{2}$, whereas the dual basis is given by $\zeta_1=H_2$, $\zeta_2=H_4$, $\zeta_3=H_1+H_6$ and $\zeta_4=H_3+H_5$. Taking account that the elements $H_i$ are related with the duals $\eta_i$ of the fundamental weights by $$\left[H_i\right]=\left[\begin{array}{cccccc} 4/3 & 1 & 5/3 & 2 & 4/3 & 2/3\\ 1 & 2 & 2 & 3 & 2 & 1\\ 5/3 & 2 & 10/3 & 4 & 8/3 & 4/3\\ 2 & 3 & 4 & 6 & 4 & 2\\ 4/3 & 2 & 8/3 & 4 & 10/3 & 5/3\\ 2/3 & 1 & 4/3 & 2 & 5/3 & 4/3 \end{array}\right]\left[\eta_i\right],$$ we see that the elements $\zeta_i$ are in the integer lattice $\mathfrak{I}'(\tilde E_6)\subset \mathfrak{I}'( E_6)$, where $\tilde E_6$ is the compact simply connected Lie group with Lie algebra $\mathfrak{e}_6$, which has centre $\mathbb{Z}_3$, and $E_6$ is the adjoint group $\tilde E_6/\mathbb{Z}_3$.

Taking into account Proposition \[concomp\], we can identify $N$ with the connected component $P^{\sigma_\varrho}_{\zeta_i}=\exp(\pi\zeta_i)P_e^{\sigma_{\varrho,i}}$, which is a totally geodesic submanifold of $G$, via $$\begin{aligned}
\label{tttt}
 g\cdot x_0\in N\mapsto\exp(\pi\zeta_i)g\sigma_{\varrho,i}(g^{-1})\in P^{\sigma_\varrho}_{\zeta_i}.\end{aligned}$$ By Theorem \[tinva\], each harmonic map $\varphi:S^2\to N\cong P^{{\sigma_\varrho}}_{\zeta_i}$ admits a $T_{\sigma_\varrho}$-invariant extended solution with values, off a discrete set, in some unstable manifold $U_\xi(G)$, with $\xi\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}_I^\varrho$. By Theorem \[nor\], this extended solution can be multiplied on the left by a constant loop in order to get a normalized extended solution with values in some unstable manifold $U_\zeta(G)$ for some ${\varrho}$-canonical element $\zeta$. Hence, if $G$ has trivial centre, the Bruhat decomposition of $\Omega_{\mathrm{alg}}G$ gives rise to $2^k$ classes of harmonic maps into $P^{\sigma_\varrho}$, that is $2^k$ classes of harmonic maps into *all* outer symmetric $G$-spaces.

However, the normalization procedure given by Theorem \[nor\] does not preserve $T_{\sigma_\varrho}$-invariance, and consequently, as we will see next, normalized extended solutions with values in the same unstable manifold $U_\zeta(G)$, for some $\varrho$-canonical element $\zeta$, correspond in general to harmonic maps into different outer symmetric $G$-spaces. Hence the classification of harmonic two-spheres into outer symmetric $G$-spaces in terms of $\varrho$-canonical elements is manifestly unsatisfactory since it does not distinguishes the underlying symmetric space. In the following sections we overcome this weakness by establishing a classification of all such harmonic maps in terms of pairs $(\zeta,\sigma)$, where $\zeta$ is a ${\varrho}$-canonical element and $\sigma$ an outer involution of $G$.

### Normalization of $T_\sigma$-invariant extended solutions {#normprocedure}

Let $\sigma$ be an outer involution of $G$. The fibre bundle morphisms $\mathcal{U}_{\xi,\xi'}$ preserve $T_{\sigma}$-invariance:

\[proposition\] If $\xi\preceq \xi'$ and $\xi,\xi'\in  \mathfrak{I}'({G})\cap \mathfrak{k}_\sigma$, then $\mathcal{U}_{\xi,\xi'}(U_\xi^{\sigma}(G))\subset U_{\xi'}^{\sigma}(G)$.

For $\Phi\in U^{\sigma}_\xi(G)$, write $\Phi=\Psi\cdot \gamma_\xi$ for some $\Psi\in \Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}$. If $\Phi$ is $T_\sigma$-invariant we have $\Psi(\lambda)\cdot \gamma_\xi=\sigma(\Psi(-\lambda))\cdot \gamma_\xi.$ Consequently, we also have $\Psi(\lambda)\cdot \gamma_{\xi'}=\sigma(\Psi(-\lambda))\cdot \gamma_{\xi'},$ which means in turn that $\mathcal{U}_{\xi,\xi'}(\Phi)=\Psi\cdot \gamma_\xi'$ is $T_{\sigma}$-invariant.

Hence, if $\Phi:S^2\setminus D\to U^{\sigma}_\xi(G)$ is an extended solution and $\xi\preceq\xi'$, with $\xi,\xi'\in\mathfrak{I}'(G)\cap \mathfrak{k}_{_\varrho}$, by Theorem \[nor\] and Proposition \[proposition\] we know that $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is a constant $T_{\sigma}$-invariant loop if ${\mathfrak{g}}_0^\xi={\mathfrak{g}}_0^{\xi'}$. However, in general, the product $\gamma \Phi$ is not $T_{\sigma}$-invariant.

\[poca\] Assume that $\gamma^{-1},\Phi\in \Omega^{\sigma} G$ and $\gamma(-1)\in P_{\xi}^{\sigma}$ for some $\xi\in\mathfrak{I}(G)\cap \mathfrak{k}_\sigma$. Take $h\in G$ such that $\gamma(-1)=h^{-1}\cdot_\sigma \exp(\pi\xi)$. Then $h\gamma\Phi h^{-1}\in\Omega^{\tau} G$, with $\tau=\mathrm{Ad}(\exp \pi\xi)\circ \sigma$.

Since $\gamma^{-1},\Phi\in \Omega^\sigma G$, a simple computation shows that $T_{\sigma}(\gamma \Phi)=\gamma(-1)^{-1}\gamma\Phi\gamma(-1).$ Since $\gamma(-1)\in P_{\xi}^{\sigma}$, there exists $h\in G$ such that $\gamma(-1)=h^{-1}\cdot_\sigma \exp(\pi\xi)=h^{-1}\exp(\pi\xi)\sigma(h)$. One can check now that $T_\tau(h\gamma \Phi h^{-1})=h\gamma \Phi h^{-1}$.

\[norminf\] Take $\xi,\xi '\in \mathfrak{I}'({G})\cap \mathfrak{k}_\sigma$ such that $\xi\preceq \xi'$. Let $\Phi:S^2\setminus D\to U^{\sigma}_\xi(G)$ be a $T_{\sigma}$-invariant extended solution. If $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is a constant loop, there exists $h\in G$ such that $\tilde{\Phi}:=h\gamma\Phi h^{-1}$ takes values in $U_{\xi'}^\tau(G)$, with $\tau=\mathrm{Ad}(\exp\pi(\xi-\xi'))\circ {\sigma}.$

Additionally, if $\sigma$ is the fundamental outer involution $\sigma_\varrho$, the harmonic map $\Phi_{-1}$ takes values in $P_\xi^{\sigma}$ and $\tilde{\Phi}_{-1}$ takes values in $P_{\xi'}^{\tau}$, which implies that $\Phi_{-1}$ is given, up to isometry, by $$\exp(\pi(\xi-\xi'))\tilde{\Phi}_{-1}:S^2\to P_\xi^\sigma.$$

Assume that $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)=\Psi\cdot \gamma_{\xi-\xi'}$ is a constant loop. We can write $\Psi\gamma_{\xi-\xi'}=\gamma^{-1} b$ for some $b:S^2\setminus D\to \Lambda^+_{\mathrm{alg}}G$. Then $$\Phi =\Psi\cdot\gamma_\xi =\Psi\cdot\gamma_{\xi-\xi'}\gamma_{\xi'}= \gamma^{-1} b\cdot\gamma_{\xi'},$$ which implies that $\gamma\Phi$ takes values in $U_{\xi'}(G)$. On the other hand, since $\gamma^{-1}$ is $T_\sigma$-invariant (by Proposition \[proposition\]), $\gamma(-1)\in P^{\sigma}$.

Take $\eta\in \mathfrak{I}'(G^\sigma)$ and $h\in G$ such that $\gamma(-1)\in P^{\sigma}_{\eta}$ and $\gamma(-1)=h^{-1}\cdot_\sigma\exp\pi\eta$. From Lemma \[poca\], we see that $\tilde\Phi:= h\gamma\Phi h^{-1}$ is $T_\tau$-invariant. Hence $\tilde{\Phi}$ takes values in $U_{\xi'}^\tau(G)$. Since $\gamma$ is constant, $\tilde\Phi$ is an extended solution.

If $\sigma=\sigma_\varrho$, then $\mathfrak{I}'(G^{\sigma_\varrho})=\mathfrak{I}'(G)\cap \mathfrak{k}_{\sigma_\varrho}$, which implies that $\eta=\xi-\xi'$. The element $h\in G$ is such that $$\gamma(-1)=h^{-1}\exp(\pi(\xi-\xi'))\sigma_\varrho(h).$$ On the other hand, since, by Theorem \[tinva\], $\Phi_{-1}$ takes values in $P_\xi^{\sigma_\varrho}$, we also have $\Phi_{-1}=g\exp(\pi\xi)\sigma_\varrho(g^{-1})$ for some lift $g:S^2\to G$. Hence $$\begin{aligned}
 \tilde\Phi_{-1}&=h\gamma(-1)\Phi_{-1} h^{-1}=\exp(\pi(\xi-\xi'))\sigma_\varrho(h)g\exp(\pi\xi)\sigma_\varrho(\sigma_\varrho(h)g)^{-1}\\&=\exp(\pi(\xi-\xi'))(\sigma_\varrho(h)g\cdot_{\sigma_\varrho} \exp\pi\xi)
 \end{aligned}$$ Hence, in view of Proposition \[concomp\], $\tilde\Phi_{-1}$ takes values in $P_{\xi'}^\tau=\exp(\pi(\xi-\xi'))P_{\xi}^\sigma$.

Under some conditions on $\xi\preceq \xi'$, the morphism $\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is always a constant loop.

\[norm2\] Take $\xi,\xi '\in \mathfrak{I}'({G})\cap \mathfrak{k}_\sigma$ such that $\xi\preceq \xi'$. Assume that $$\begin{aligned}
 {\mathfrak{g}}_{2i}^\xi\cap \mathfrak{m}_\sigma^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i}{\mathfrak{g}}^{\xi-\xi'}_j,\quad\quad
 {\mathfrak{g}}_{2i-1}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i-1}{\mathfrak{g}}^{\xi-\xi'}_j,\label{toing2}
 \end{aligned}$$ for all $i> 0$. Then, $\mathcal{U}_{\xi,\xi-\xi'}:U_\xi^\sigma(G)\to U_{\xi-\xi'}^\sigma(G)$ transforms $T_\sigma$-invariant extended solutions in constant loops.

Given an extended solution $\Phi:S^2\setminus D\to U^\sigma_\xi(G)$, choose $\Psi:S^2\setminus D \to \Lambda_{\mathrm{alg}}^+G^{\mathbb{C}}$ such that $\Phi=\Psi\cdot\gamma_\xi $ and $T_\sigma(\Psi)=\Psi$. Differentiating this we see that $$\label{ondepsi}
\mathrm{Im}\Psi^{-1}\Psi_z\subset \bigoplus_{i\geq 0}\lambda^{2i}\mathfrak{k}_\sigma^{\mathbb{C}}\oplus\bigoplus_{i\geq 0}\lambda^{2i+1}\mathfrak{m}_\sigma^{\mathbb{C}}.$$ Write $\Psi^{-1}\Psi_z=\sum_{r\geq 0}\lambda^rX'_r$. Since $\xi\preceq \xi-\xi'$, by Proposition \[popo\] and Proposition \[proposition\], $\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is an extended solution with values in $U^\sigma_{\xi-\xi'}(G)$. Hence, taking into account Lemma \[poi\], in order to prove that $\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is constant we only have to check that the component of $X'_r$ over ${\mathfrak{g}}^{\xi-\xi'}_{r+1}$ vanishes for all $r\geq 0$.

From and we see that, for $r=2i$, $X'_{2i}$ takes values in $\bigoplus_{j\leq 2i+1}  {\mathfrak{g}}_{j}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}$. But, since $\xi\preceq \xi-\xi'$ and, by hypothesis, holds, we have $$\bigoplus_{j\leq 2i+1}  {\mathfrak{g}}_{j}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}=\big(\bigoplus_{j\leq 2i}  {\mathfrak{g}}_{j}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}\big)\oplus \big({\mathfrak{g}}_{2i+1}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}\big)\subset
\big(\bigoplus_{j\leq 2i}  {\mathfrak{g}}_{j}^{\xi-\xi'}\cap \mathfrak{k}_\sigma^{\mathbb{C}}\big)\oplus  \bigoplus_{0\leq j<2i+1}{\mathfrak{g}}^{\xi-\xi'}_j.$$ Hence the component of $X'_{2i}$ over ${\mathfrak{g}}^{\xi-\xi'}_{2i+1}$ vanishes for all $i\geq 0$. Similarly, for $r=2i-1$, $X'_{2i-1}$ takes values in $\bigoplus_{j\leq 2i}  {\mathfrak{g}}_{j}^\xi\cap\mathfrak{m}_\sigma^{\mathbb{C}}$, and we can check that the component of $X'_{2i-1}$ over ${\mathfrak{g}}^{\xi-\xi'}_{2i}$ vanishes for all $i>0$.

Hence $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)=\Psi\cdot \gamma_{\xi-\xi'}$ is a constant loop.

We say that $\zeta\in  \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$ is a $\varrho$-*semi-canonical* element if $\zeta$ is of the form $\zeta=\sum_{i\in I}n_i\zeta_i$ with $1\leq n_i\leq 2m_i$, where $m_i$ is the least positive integer which makes $m_i\zeta_i\in\mathfrak{I}'({G}^{\sigma_\varrho})$.

\[crorol\] Take $\xi\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$, with $I\subset\{1,\ldots,k\}$. Let $\Phi:S^2\setminus D\to U^{\sigma_\varrho}_\xi(G)$ be a $T_{\sigma_\varrho}$-invariant extended solution, and let $\varphi:S^2\to P_\xi^{\sigma_\varrho}$ be the corresponding harmonic map. Then there exist $h\in G$, a constant loop $\gamma$, and a $\varrho$-*semi-canonical* $\zeta$ such that $\tilde\Phi:=h\gamma\Phi h^{-1}$ defined on $S^2\setminus D$ takes values in $U^{\sigma_\varrho}_{\zeta}(G)$. The harmonic map $\tilde\Phi_{-1}$ takes values in $P^{\sigma_\varrho}_{\zeta}= P^{\sigma_\varrho}_{\xi}$ and coincides with $\varphi$ up to isometry.

Write $\xi=\sum_{i\in I}r_i\zeta_i$, with $r_i>0$. For each $i\in I$, let $n_i$ be the unique integer number in $\{1,\ldots, 2m_i\}$ such that $n_i=r_i\mod 2m_i.$ Set $\zeta=\sum_{i\in I}n_i\zeta_i$. It is clear that $\xi\preceq \zeta$ and $\zeta \in  \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$. Observe also that conditions hold automatically for any $\xi'\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$ satisfying $\xi\preceq\xi'$. In particular they hold for $\xi'=\zeta$. Finally, since $\xi-\zeta=2\sum_{i\in I}m_ik_i\zeta_i$ for some nonnegative integer numbers $k_i$, then $\exp{\pi(\xi-\zeta)}=e$, and the result follows from Propositions \[norminf\] and \[norm2\].

### Classification of harmonic two-spheres into outer symmetric spaces {#classs}

To sum up, in order to classify all harmonic two-spheres into outer symmetric spaces we proceed as follows:

1.  Start with a fundamental outer involution $\sigma_\varrho$ and let $N$ be an outer symmetric $G$-space corresponding to an involution of the form $\sigma_\varrho$ or $\sigma_{\varrho,i}$ of $G$, according to , where the element $\zeta_i$ is in the conditions of Theorem \[murak\]. We assume that $\exp2\pi\zeta_i=e$, that is $\zeta_i\in \mathfrak{I}'({G}^{\sigma_\varrho})$. Let $\varphi:S^2\to N$ be an harmonic map and identify $N$ with $P_{\zeta_i}^{\sigma_\varrho}=\exp(\pi\zeta_i)P_e^{\sigma_{\varrho,i}}$ via the totally geodesic embedding . If $N$ is the fundamental outer space with involution $\sigma_\varrho$ we simply identify $N$ with $P_e^{\sigma_\varrho}$ via $\iota_{\sigma_\varrho}$.

2.  By Theorem \[tinva\], $\varphi:S^2\to N\cong P_{\zeta_i}^{\sigma_\varrho}$ admits a $T_{\sigma_\varrho}$-invariant extended solution $\Phi:S^2\to\Omega^{\sigma_\varrho} G$ which takes values, off some discrete subset $D$, in some unstable manifold $U_{\zeta'}^{\sigma_\varrho}(G)$, with $\zeta'\in\mathfrak{I}'({G}^{\sigma_\varrho})$; moreover, $P_{\zeta'}^{\sigma_\varrho}=P_{\zeta_i}^{\sigma_\varrho}$.

3.  By Corollary \[crorol\], we can assume that $\zeta'$ is a $\varrho$-semi-canonical element in $\mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$. If $\zeta$ is a $\varrho$-canonical element such that $\zeta'\preceq \zeta$ and $\mathcal{U}_{\zeta',\zeta'-\zeta}(\Phi)$ is constant, then, taking into account Proposition \[norminf\], there exists a $T_\tau$-invariant extended solution $\tilde\Phi: S^2\setminus D\to U^\tau_{\zeta}(G),$ where $$\label{otau}\tau=\mathrm{Ad}(\exp \pi(\zeta'-\zeta))\circ \sigma_\varrho,$$ such that the harmonic map $\varphi$ is given, up to isometry, by $\tilde\Phi_{-1}:S^2\to P_{\zeta}^\tau.$ Here we identify $N$ with $P_{\zeta}^\tau =\exp(\pi(\zeta'-\zeta)) P_{\zeta_i}^{\sigma_\varrho}$ via the composition of with the left multiplication by $\exp(\pi(\zeta'-\zeta))$.

4.  By Proposition \[norm2\], there always exists a $\varrho$-canonical element $\zeta$ in such conditions.

Hence, we classify harmonic spheres into outer symmetric $G$-spaces in terms of pairs $(\zeta,\tau)$, where $\zeta$ is a $\varrho$-canonical element and $\tau$ is an outer involution of the form for some $\varrho$-semi-canonical element $\zeta'$ with $\zeta'\preceq \zeta$.

### Weierstrass Representation for $T_\sigma$-invariant Extended Solutions

From and Proposition \[BG\], we obtain the following.

\[sigmaweirstrass\] Let $\Phi:M\to \Omega_{\mathrm{alg}}^\sigma G$ be an extended solution. There exists a discrete set $D'\supseteq D$ of $M$ such that $\Phi{\big|_{M\setminus D'}}=\exp C\cdot \gamma_\xi$ for some holomorphic vector-valued function $C: M\setminus D'\to (\mathfrak{u}^0_\xi)_\sigma$, where $(\mathfrak{u}^0_\xi)_\sigma$ is the finite dimensional nilpotent subalgebra of $\Lambda^+_{\mathrm{alg}}\mathfrak g^{\mathbb{C}}$ defined by $$(\mathfrak{u}^0_\xi)_\sigma=\bigoplus_{0\leq 2i<r(\xi)}\lambda^{2i}(\mathfrak{p}^\xi_{2i})^\perp\cap \mathfrak{k}^{\mathbb{C}}_\sigma\oplus \bigoplus_{0\leq 2i+1<r(\xi)}\lambda^{2i+1}(\mathfrak{p}^\xi_{2i+1})^\perp\cap \mathfrak{m}^{\mathbb{C}}_\sigma,$$ with $(\mathfrak{p}^\xi_i)^\perp=\bigoplus _{i<j\leq r(\xi)}\mathfrak{g}_j^\xi$. Moreover, $C$ can be extended meromorphically to $M$.

Examples
========

Next we will describe explicit examples of harmonic spheres into *classical* outer symmetric spaces.

Outer symmetric $SO(2n)$-spaces
-------------------------------

For details on the structure of $\mathfrak{so}(2n)$ see [@fulton_harris]. Consider on $\mathbb{R}^{2n}$ the standard inner product $\langle \cdot, \cdot \rangle$ and fix a complex basis $\mathbf{u}=\{u_1,\ldots,u_n,\overline{u}_1,\ldots,\overline{u}_n\}$ of $\mathbb{C}^{2n}=(\mathbb{R}^{2n})^\mathbb{C}$ satisfying $$\label{us}
  \langle u_i,u_j\rangle=0, \quad  \langle u_i,\overline{u}_j\rangle=\delta_{ij},\quad \mbox{for all $1\leq i,j\leq n$}.$$ Throughout this section we will denote by $V_l$ the $l$-dimensional isotropic subspace spanned by $\overline{u}_1,\ldots,\overline{u}_l$.

Set $E_i=E_{i,i}-E_{n+i,n+i}$, where $E_{j,j}$ is a square matrix, with respect to the basis $\mathbf{u}$, whose $(j,j)$-entry is $\mathrm{i}$ and all other entries are $0$. The complexification ${\mathfrak{t}}^{\mathbb{C}}$ of the algebra ${\mathfrak{t}}$ of diagonal matrices $\sum a_iE_i$, with $a_i\in \mathbb{R}$ and $\sum a_i=0$, is a Cartan subalgebra of $\mathfrak{so}(2n)^{\mathbb{C}}$. Let $\{L_1,\ldots,L_n\}$ be the dual basis in $\mathrm{i}{\mathfrak{t}}^*$ of $\{E_1,\ldots,E_n\}$, that is $L_i(E_j)=\mathrm{i}\delta_{ij}$. The roots of $\mathfrak{so}(2n)$ are the vectors $\pm L_i\pm L_j$ and $\pm L_i\mp L_j$, with $i\neq j$ and $1\leq i,j\leq n$.

Consider the endomorphisms $$\label{geradores}
 X_{i,j}=E_{i,j}-E_{n+j,n+i},\,\,Y_{i,j}=E_{i,n+j}-E_{j,n+i},\,\, Z_{i,j}=E_{n+i,j}-E_{n+j,i},$$ where $E_{i,j}$, with $i\neq j$, is a square matrix whose $(i,j)$-entry is $1$ and all other entries are $0$. The root spaces of $L_i-L_j$, $L_i+L_j$ and $-L_i-L_j$, respectively, are generated by the endomorphisms $X_{i,j}$, $Y_{i,j}$ and $Z_{i,j}$, respectively.

Fix the positive root system $\Delta^+=\{L_i\pm L_j\}_{i<j}.$ The positive simple roots are $\alpha_i=L_i-L_{i+1}$, for $1\leq i\leq n-1$, and $\alpha_n=L_{n-1}+L_n$. The vectors of the dual basis $\{H_1,\ldots,H_n\}\subset {\mathfrak{t}}$ are given by $H_i  =E_1+E_2+\ldots+E_i$, for $1\leq i\leq n-2$, $$\mbox{$H_{n-1}  =\frac12(E_1+E_2+\ldots+E_{n-1}-E_n)$, and $H_n  =\frac12(E_1+E_2+\ldots+E_{n-1}+E_n)$}.$$

Consider the non-trivial involution $\varrho$ of the corresponding Dynkin diagram,

(-0.9,1.11)(4.85,2.76) (0,0)(1,0) (1,0)(2,0) (2,0)(3,0) (3,0)(4,0) (2,0)(1.95,0.21) (-0.6,1.86)[$\alpha_1$]{} (0.43,1.84)[$\alpha_2$]{} (0.92,-0.13)[$\alpha_3$]{} (1.8,1.86)[$\alpha_{n-3}$]{} (2.7,1.86)[$ \alpha_{n-2}$]{} (3.97,-0.13)[$\alpha_6$]{} (-0.5,2)(0.5,2) (3,2)(3.86,1.5) (3,2)(3.85,2.5) (2,2)(3,2) (0.84,2)[$\ldots\ldots\ldots$]{} (3.96,1.55)[$\alpha_n$]{} (3.96,2.63)[$\alpha_{n-1}$]{}

(0,0) (1,0) (2,0) (3,0) (4,0) (1.95,0.21) (0,4) (-0.5,2) (0.5,2) (3,2) (3.86,1.5) (3.85,2.5) (-2.26,0.91) (2,2)

This involution fixes $\alpha_i$ if $i\leq n-2$ and $\varrho(\alpha_{n-1})= \alpha_n$. The corresponding semi-fundamental basis $\pi_{\mathfrak{k}_\varrho}(\Delta_0)=\{\beta_1,\ldots,\beta_{n-1}\}$ is given by $$\mbox{$\beta_i=\alpha_i=L_i-L_{i+1}$, if $i\leq n-2$, and $\beta_{n-1}=\frac12(\alpha_{n-1}+\alpha_{n})=L_{n-1}$},$$ whereas the dual basis $\{\zeta_1,\ldots,\zeta_{n-1}\} $ is given by $\zeta_i=E_1+\ldots+E_i$, with $i=1,\ldots,n-1$. Since each $\zeta_i$ belongs to the integer lattice $\mathfrak{I}(SO(2n)^{\sigma_\varrho})$, we have:

The $\varrho$-semi-canonical elements of $SO(2n)$ are precisely the elements $\zeta=\sum_{i=1}^{n-1} m_i\zeta_i$ such that $m_i\in\{0,1,2\}$ for $1\leq i\leq n-1$.

The fundamental outer symmetric $SO(2n)$-space is the real projective space $\mathbb{R}P^{2n-1}$, and the associated outer symmetric $SO(2n)$-spaces are the real Grassmannians $G_p(\mathbb{R}^{2n})$ with $p>1$ odd.

### Harmonic maps into real projective spaces $\mathbb{R}P^{2n-1}$.

Consider as base point the one dimensional real vector space $V_0$ spanned by $e_n=(u_n+\overline{u}_n)/\sqrt{2}$ in $\mathbb{R}^{2n}$, which establishes an identification of $\mathbb{R}P^{2n-1}$ with $SO(2n)/S(O(1)O(2n-1)).$ Denote by $\pi_{V_0}$ and $\pi_{V_0}^\perp$ the orthogonal projections onto $V_0$ and $V_0^\perp$, respectively. The fundamental involution is given by $\sigma_\varrho=\mathrm{Ad}(s_0)$, where $s_0=\pi_{V_0}-\pi_{V_0}^\perp$. Following the classification procedure established in Section \[classs\], we start by identifying $\mathbb{R}P^{2n-1}$ with $P_e^{\sigma_\varrho}$.

\[classesproj\] Each harmonic map $\varphi:S^2\to\mathbb{R}P^{2n-1}$ belongs to one of the following classes: $(\zeta_l,\sigma_{\varrho,l})$, with $1\leq l\leq n-1$.

Let $\zeta$ be a $\varrho$-semi-canonical element and write $$\label{ozeta}\zeta=\sum_{i\in I_1}\zeta_i+ \sum_{i\in I_2}2\zeta_i$$ for some disjoint subsets $I_1$ and $I_2$ of $\{1,\ldots,n-1\}$. By Proposition \[fundcan\], $P^{\sigma_\varrho}_\zeta\cong \mathbb{R}P^{2n-1}$ if and only if either $I_1=\emptyset$ or $I_1=\{n-1\}$. Suppose that $I_1=\{n-1\}$. In this case, $\exp\pi\zeta=\exp\pi\zeta_{n-1}  \in P_{\zeta_{n-1}}^{\sigma_\varrho}$. We claim that $P_{\zeta_{n-1}}^{\sigma_\varrho}$ is not the connected component of $P^{\sigma_\varrho}$ containing the identity $e$. Write $\exp\pi\zeta_{n-1}=\pi_V-\pi_V^\perp,$ where $V$ is the two-dimensional real space spanned by $e_{n}$ and $e_{2n}$. For each $g\in  P_e^{\sigma_\varrho}$, since the $G$-action $\cdot_{\sigma_\varrho}$ defined by is transitive, we have $g=h\cdot_{\sigma_\varrho}e=hs_0h^{-1}s_0$ for some $h\in G$, which means that $gs_0=hs_0h^{-1}$. In particular, the $+1$-eigenspaces of $gs_0$ must be $1$-dimensional. However, a simple computation shows that the $+1$-eigenspace of $\exp(\pi\zeta_{n-1})s_0$ is $3$-dimensional, which establishes our claim.

Then, any harmonic map $\varphi:S^2\to\mathbb{R}P^{2n-1}\cong P_e^{\sigma_\varrho}$ admits a $T_{\sigma_\varrho}$-invariant extended solution $\Phi:S^2\setminus D\to U^{\sigma_\varrho}_\zeta(SO(2n))$ with $\zeta$ a $\varrho$-semi-canonical element of the form $\zeta=\sum_{i\in I_2}2\zeta_i.$ Set $l=\max I_2$. Next we check that $\zeta$ and $\zeta_l$ satisfy the conditions of Proposition \[norm2\], with $\xi=\zeta$ and $\xi'=\zeta_l$. It is clear that $\zeta\preceq \zeta_l$. Now, according to and , we can take $\Delta'_\varrho=\{L_i-L_n, L_n-L_i\}$. Hence, for $i>0$, $$\mathfrak{g}_{2i}^\zeta\cap \mathfrak{m}_\varrho^{\mathbb{C}}=\bigoplus_{\alpha\in \Delta'_\varrho\cap \Delta_\zeta^{2i}}({\mathfrak{g}}_\alpha\oplus {\mathfrak{g}}_{\varrho(\alpha)})\cap\mathfrak{m}^{\mathbb{C}}_\varrho,$$ where $\Delta_\zeta^{2i}=\{\alpha\in \Delta|\, \alpha(\zeta)=2i\mathrm{i}\}$. Since $$(L_j-L_n)(\zeta)=(\alpha_j+\alpha_{j+1}+\ldots+\alpha_{n-1})(\zeta)=2|I_2\cap \{j,\ldots,n-1\}|\mathrm{i},$$ we have $$\Delta'_\varrho\cap \Delta_\zeta^{2i}=\{L_j-L_n|\,\, \mbox{$1\leq j\leq l$, and  $|I_2\cap\{j,\ldots,l\}|=i$}\}.$$ Then, given a root $\alpha=L_j-L_n\in \Delta'_\varrho\cap \Delta_\zeta^{2i}$ (in particular, $j\leq l$) we have $\alpha(\zeta-\zeta_l)=(2i-1)\mathrm{i},$ which means that ${\mathfrak{g}}_\alpha\subset {\mathfrak{g}}^{\zeta-\zeta_l}_{2i-1}$. Consequently, $${\mathfrak{g}}_{2i}^\zeta\cap \mathfrak{m}_\varrho^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i}{\mathfrak{g}}^{\zeta-\zeta_l}_j.$$ Since ${\mathfrak{g}}_{2i-1}^\zeta=\{0\}$ for all $i$, we conclude that holds, and the statement follows from Propositions \[norminf\] and \[norm2\].

It is known [@calabi_1967] that there are no full harmonic maps $\varphi:S^2\to\mathbb{R}P^{2n-1}$. The class of harmonic maps associated to $(\zeta_l,\sigma_{\varrho,l})$ consists precisely of those $\varphi$ with $\varphi(S^2)$ contained, up to isometry, in some $\mathbb{R}P^{2l}$, as shown in the next theorem.

\[RAs\] Given $1\leq l\leq n-1$, any harmonic map $\varphi:S^2\to\mathbb{R}P^{2n-1}$ in the class $(\zeta_l,\sigma_{\varrho,l})$ is given by $$\label{projspace}
  \varphi={R}\cap(A\oplus \overline{A})^\perp,$$ where ${R}$ is a constant $2l+1$-dimensional subspace of $\mathbb{R}^{2n}$ and $A$ is a holomorphic isotropic subbundle of $S^2\times {R}$ of rank $l$ satisfying $\partial A\subseteq \overline{A}^\perp$. The corresponding extended solutions have uniton number $2$ with respect to the standard representation of $SO(2n)$.

Let $\varphi:S^2\to\mathbb{R}P^{2n-1}$ be a harmonic map in the class $(\zeta_l,\sigma_{\varrho,l})$. This means that $\varphi$ admits an extended solution $\Phi:S^2\setminus D \to U_{\zeta_l}^{\sigma_{\varrho,l}}(SO(2n))$. Up to isometry, $\varphi$ is given by $\Phi_{-1}$, which takes values in $P_{\zeta_l}^{\sigma_{\varrho,l}}=\exp(\pi \zeta_l)P_e^{\sigma_{\varrho}}$. This connected component is identified with $\mathbb{R}P^{2n-1}$ via $$\label{identi}
 g\cdot V_0\mapsto \exp(\pi \zeta_l)g\sigma_\varrho(g^{-1}).$$ Write $\gamma_{\zeta_l}(\lambda)=\lambda^{-1}\pi_{V_l}+\pi_{V_l\oplus\overline{V}_l}^\perp+\lambda \pi_{\overline{V}_l},$ where $V_l$ is the $l$-dimensional isotropic subspace spanned by $\overline{u}_1,\ldots,\overline{u}_l$.

We have $r(\zeta_l)=2$ if $l>1$ and $r(\zeta_1)=1$. Consequently, by Theorem \[sigmaweirstrass\], $$(\mathfrak{u}^0_{\zeta_l})_{\sigma_{\varrho,l}}=(\mathfrak{p}^{\zeta_l}_{0})^\perp\cap \mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,l}}\oplus \lambda (\mathfrak{p}^{\zeta_l}_{1})^\perp\cap \mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,l}}.$$ Here $(\mathfrak{p}^{\zeta_l}_{1})^\perp={\mathfrak{g}}_2^{\zeta_l}$, which is the null space for $l=1$. For $l>1$, since $\zeta_l=E_1+\ldots+E_l$, we have ${\mathfrak{g}}_2^{\zeta_l}=\{L_i+L_j|\, 1\leq i< j\leq l\}\subset \Delta(\mathfrak{k}_\varrho)$ and, from , $$\mathfrak{m}_{\sigma_{\varrho,l}}^{\mathbb{C}}=\bigoplus \mathfrak{g}^{\zeta_l}_{2i+1}\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}\oplus\bigoplus \mathfrak{g}^{\zeta_l}_{2i}\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}.$$ Hence $(\mathfrak{p}^{\zeta_l}_{1})^\perp\cap \mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,l}}= {\mathfrak{g}}_2^{\zeta_l}\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}=\{0\}.$ Then, for any $l\geq 1$, we can write $\Phi=\exp C\cdot \gamma_{\zeta_l}$ for some holomorphic function $$C:S^2\setminus D\to (\mathfrak{p}^{\zeta_l}_{0})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,l}} =({\mathfrak{g}}_1^{\zeta_l}\oplus{\mathfrak{g}}_2^{\zeta_l})\cap \mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,l}},$$ which means that $\Phi$ is a $S^1$-invariant extended solution with uniton number $2$: $$\label{phiw}
\Phi_\lambda=\lambda^{-1}\pi_{W}+\pi_{W\oplus\overline{W}}^\perp+\lambda \pi_{\overline{W}},$$ where $W$ is a holomorphic isotropic subbundle of $S^2\times \mathbb{R}^{2n}$ of rank $l$ satisfying the superhorizontality condition $\partial W\subseteq \overline{W}^\perp$.

Set $\tilde{V}_l= V_l\oplus \overline{V}_l$ and $\tilde{W}= W\oplus \overline{W}$. The $T_{{\sigma_{\varrho,l}}}$-invariance of $\Phi$ implies that $$\label{inva} [\pi_W,\pi_{V_0\oplus \tilde V_l}]=0.$$ Now, write $\varphi=g\cdot V_0$ and consider the identification . We must have $$\label{phirp}
  \Phi_{-1}= \exp(\pi \zeta_l)g\sigma_\varrho(g^{-1})=\exp(\pi \zeta_l)(\pi_\varphi-\pi_\varphi^\perp)s_0.$$ From and we obtain $$\label{pias}
\pi_\varphi-\pi_\varphi^\perp=\mathrm{Ad}(s_0)\big(\pi_{V_0\oplus \tilde V_l}\pi_{\tilde W}^\perp+\pi_{V_0\oplus \tilde V_l}^\perp\pi_{\tilde W}
-\pi_{V_0\oplus \tilde V_l}\pi_{\tilde W}-\pi_{V_0\oplus \tilde V_l}^\perp\pi_{\tilde W}^\perp\big).$$ In view of , we see that $\pi_{V_0\oplus \tilde V_l}\pi_{\tilde W}^\perp+\pi_{V_0\oplus \tilde V_l}^\perp\pi_{\tilde W}$ is an orthogonal projection, and implies that this must be an orthogonal projection onto a $1$-dimensional real subspace. Then, one of its two terms vanishes, that is either $\tilde W\subset V_0\oplus \tilde V_l$ or $\tilde W^\perp\subset (V_0\oplus \tilde V_l)^\perp$. For dimensional reasons, we see that the second case can not occur. Hence, we have $$\pi_\varphi=\mathrm{Ad}(s_0)( \pi_{V_0\oplus \tilde V_l}\pi_{\tilde W}^\perp)= \pi_{V_0\oplus \tilde V_l}\mathrm{Ad}(s_0)(\pi_{\tilde W}^\perp),$$ that is holds with ${R}=V_0\oplus  V_l\oplus \overline V_l$ and $A=s_0(W)$.

If $\varphi$ is full in ${R}$, then the isotropic subbundle $A$ is the $l$-osculating space of some full totally isotropic holomorphic map $f$ from $S^2$ into the complex projective space of ${R}$, the so called *directrix curve* of $\varphi$. That is, in a local system of coordinates $(U,z)$, we have $A(z)=\mathrm{Span}\big\{g,g',\ldots,g^{(l-1)}\}$, where $g$ is a lift of $f$ over $U$ and $g^{(r)}$ the $r$-th derivative of $g$ with respect to $z$. Hence, formula agrees with the classification given in Corollary 6.11 of [@eells_wood_1983].

Let us consider the case $n=2$. We have only one class of harmonic maps: $(\zeta_1,\sigma_{\varrho,1})$. From Theorem \[RAs\], any such harmonic map $\varphi:S^2\to{\mathbb{R}}P^3$ is given by $\varphi=R\cap(A\oplus\overline{A})^\perp$, where $R$ is a constant 3-dimensional subspace of ${\mathbb{R}}^4$ and $A$ a holomorphic isotropic subbundle of $S^2\times R$ of rank 1 such that $\partial A\subseteq\overline{A}^\perp$. Taking into account Theorem \[sigmaweirstrass\], any such holomorphic subbundles $A$ can be obtained from a meromorphic function $a$ on $S^2$ as follows.

We have $\zeta_1=E_1$ and the corresponding extended solutions have uniton number $r(\zeta_1)=1$ (with respect to the standard representation). Any extended solution $\Phi:S^2\setminus D\to U_{\zeta_1}^{\sigma_{\varrho,1}}(SO(4))$ is given by $\Phi=\exp C\cdot\gamma_{\zeta_1}$, with $\gamma_{\zeta_1}(\lambda)=\lambda^{-1}\pi_{{V}_1}+\pi_{V_1\oplus\overline{V}_1}^\perp+\lambda\pi_{\overline{V}_1}$, for some holomorphic vector-valued function $C:S^2\setminus D\to(\mathfrak{u}_{\zeta_1}^0)_{\sigma_{\varrho,1}}$, where $$(\mathfrak{u}_{\zeta_1}^0)_{\sigma_{\varrho,1}}=(\mathfrak{p}_0^{\zeta_1})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}=\mathfrak{g}_1^{\zeta_1}\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}
=(\mathfrak{g}_{L_1-L_2}\oplus\mathfrak{g}_{L_1+L_2})\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}.$$

Considering the root vectors $X_{i,j},Y_{i,j},Z_{i,j}$ as defined in , we have $Y_{1,2}=\sigma_{\varrho,1}(X_{1,2})$. Hence $C=a(z)(X_{1,2}+Y_{1,2})$ where $a(z)$ is a meromorphic function on $S^2$. In this case, from , it follows that $(\exp C)^{-1}(\exp C)_z=C_z$, and it is clear that the extended solution condition for $\Phi$ holds independently of the choice of the meromorphic function $a(z)$. Then, with respect to the complex basis $\mathbf{u}=\{u_1,u_2,\overline{u}_1,\overline{u}_2\}$, $$\label{zeta1rp3}\exp C\cdot\gamma_{\zeta_1}=\left[\begin{array}{cccc} 1& a & -a^2 & a\\ 0 & 1 & -a & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & -a & 1 \end{array}\right]\cdot\gamma_{\zeta_1}$$ and the subbundle $A$ of $R=\mathrm{Span}\{{u}_1,\overline{u}_1, u_2+\overline{u}_2\}$ is given by $A=\exp C\cdot {V}_1=\mathrm{span}\{(a^2,a,-1,a)\}$, which satisfies $\partial A\subseteq\overline{A}^\perp$.

Any harmonic two-sphere into ${\mathbb{R}}P^5$ in the class $(\zeta_1,\sigma_{\varrho,1})$ takes values in some ${\mathbb{R}}P^3$ inside ${\mathbb{R}}P^5$ and so it is essentially of the form (\[zeta1rp3\]). Next we consider the Weierstrass representation of harmonic spheres into ${\mathbb{R}}P^5$ in the class $(\zeta_2,\sigma_{\varrho,2})$, which are given by $\varphi=R\cap(A\oplus\overline{A})^\perp$, where $R$ is a constant 5-dimensional subspace of ${\mathbb{R}}^6$ and $A$ a holomorphic isotropic subbundle of $S^2\times R$ of rank 2 such that $\partial A\subseteq\overline{A}^\perp$. We have $\zeta_2=E_1+E_2$, then $r(\zeta_2)=2$. Any extended solution $\Phi:S^2\setminus D\to U_{\zeta_2}^{\sigma_{\varrho,2}}(SO(6))$ is given by $\Phi=\exp C\cdot\gamma_{\zeta_2}$, with $\gamma_{\zeta_2}(\lambda)=\lambda^{-1}\pi_{{V}_2}+\pi_{V_2\oplus\overline{V}_2}^\perp+\lambda\pi_{\overline{V}_2},$ for some holomorphic vector-valued function $C:S^2\setminus D\to(\mathfrak{u}_{\zeta_2}^0)_{\sigma_{\varrho,2}}$, where $$\begin{aligned}
 (\mathfrak{u}_{\zeta_2}^0)_{\sigma_{\varrho,2}}  =\left((\mathfrak{g}_{L_1- L_3}\oplus \mathfrak{g}_{L_1+L_3})\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,2}}\right)\oplus\left((\mathfrak{g}_{L_2- L_3}\oplus \mathfrak{g}_{L_2+L_3})\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,2}}\right)\oplus \mathfrak{g}_{L_1+L_2}. \end{aligned}$$

We have $Y_{1,3}=\sigma_{\varrho,2}(X_{1,3})$ and $Y_{2,3}=\sigma_{\varrho,2}(X_{2,3})$. Hence we can write $$C=a(z)(X_{1,3}+Y_{1,3})+b(z)(X_{2,3}+Y_{2,3})+c(z)Y_{1,2}$$ where $a(z)$, $b(z)$ and $c(z)$ are meromorphic functions on $S^2$.

Now, $\Phi=\exp{C}\cdot \gamma_{\zeta_2}$ is an extended solution if and only if, in the expression $C_z-\frac{1}{2!}(\mathrm{ad} C)C_z,$ which does not depend on $\lambda$, the component on ${\mathfrak{g}}_2^{\zeta_2}={\mathfrak{g}}_{L_1+L_2}$ vanishes. Since $Y_{1,2}=[Y_{2,3},X_{1,3}]=[X_{2,3},Y_{1,3}]$ and $[X_{1,3},X_{2,3}]=[Y_{1,3},Y_{2,3}]=0$, this holds if and only if $c'=ba'-ab'$, where prime denotes $z$-derivative. Since $A=\exp C\cdot V_2$, we can compute $\exp C$ in order to conclude that the holomorphic subbundle $A$ of $R=\mathrm{Span}\{{u}_1,u_2,\overline{u}_1,\overline{u}_2, u_3+\overline{u}_3\}$ is given by $$A=\mathrm{Span}\{(a^2,ab+c,a,-1,0,a),(ab-c,b^2,b,0,-1,b)\}.$$

### Harmonic maps into Real Grassmanians.

Let $\zeta'$ be a $\varrho$-semi-canonical element of $SO(2n)$ given by , for some disjoint subsets $I_1$ and $I_2$ of $\{1,\ldots,n-1\}$. By Proposition \[fundcan\], we know that $P^{\sigma_\varrho}_{\zeta'}\cong \mathbb{R}P^{2n-1}$ if and only if either $I_1=\emptyset$ or $I_1=\{n-1\}$. More generally we have:

\[grassd\] If $I_1=\{i_1>i_2>\ldots>i_r\}$ and $d=\sum_{j=1}^r(-1)^{j+1}i_j$, then $P^{\sigma_\varrho}_{\zeta'}\cong G_{2d+1}(\mathbb{R}^{2n})$.

For $\zeta'$ of the form , set $\zeta'_{I_1}=\sum_{i\in I_1}\zeta_i$. Clearly, $\exp\pi\zeta'=\exp\pi\zeta'_{I_1}$, and, by Proposition \[concomp\], $P^{\sigma_\varrho}_{\zeta'}$ is a symmetric space with involution $$\tau=\mathrm{Ad}(\exp\pi\zeta'_{I_1})\circ \sigma_\varrho=\mathrm{Ad}(s_0\exp \pi \zeta'_{I_1}).$$ We have $$\zeta'_{I_1}=r(E_1+\ldots + E_{i_r})+(r-1)(E_{i_r+1}+\ldots +E_{i_{r-1}})+\ldots +(E_{i_2+1}+\ldots +E_{i_1}),$$ and consequently, with the convention $V_{i_0}=V_n$ and $V_{i_{r+1}}=\{0\}$, $$\exp\pi\zeta'_{I_1}=\sum_{j=0}^r(-1)^j\pi_{i_j-i_{j+1}}+\sum_{j=0}^r(-1)^j\overline{\pi}_{i_j-i_{j+1}},$$ where $\pi_{i_j-i_{j+1}}$ is the orthogonal projection onto $V_{i_j}\cap V_{i_{j+1}}^\perp$ and $\overline{\pi}_{i_j-i_{j+1}}$ the orthogonal projection onto the corresponding conjugate space. Hence, the $+1$-eigenspace of $s_0\exp\zeta'_{I_1} $ has dimension $2d+1$, with $d=\sum_{j=1}^r(-1)^{j+1}i_j$, which means that $P^{\sigma_\varrho}_{\zeta'}\cong G_{2d+1}(\mathbb{R}^{2n})$.

In particular, we have $P_{\zeta_d}^{\sigma_\varrho}\cong G_{2d+1}(\mathbb{R}^{2n})$ for each $d\in\{1,\ldots,n-1\}$.

\[cangrass\] Each harmonic map from $S^2$ into the real Grassmannian $G_{2d+1}(\mathbb{R}^{2n})$ belongs to one of the following classes: $(\zeta,\mathrm{Ad}\exp\pi(\tilde\zeta-\zeta)\circ \sigma_{\varrho,l}),$ where $\zeta$ and $\tilde \zeta$ are $\varrho$-canonical elements such that $\tilde\zeta\preceq \zeta$ and $\tilde\zeta=\sum_{i\in I_1}\zeta_i+\zeta_l$, where

1.  $I_1=\{i_1>i_2>\ldots>i_r\}$ satisfies $d=\sum_{j=1}^r(-1)^{j+1}i_j$;

2.  $l\in\{0,1,\ldots,n-1\}$ and $l\notin I_1$ (if $l=0$, we set $\zeta_0=0$).

We consider harmonic maps into $P_{\zeta_d}^{\sigma_\varrho}\cong G_{2d+1}(\mathbb{R}^{2n})$. Let $\zeta'$ be a $\varrho$-semi-canonical element and write $\zeta'=\sum_{i\in I_1}\zeta_i+ \sum_{i\in I_2}2\zeta_i$ for some disjoint subsets $I_1$ and $I_2$ of $\{1,\ldots,n-1\}$. By Proposition \[grassd\], $P^{\sigma_\varrho}_{\zeta'}\cong G_{2d+1}(\mathbb{R}^{2n})$ if and only if either $d=\sum_{j=1}^r(-1)^{j+1}i_j$ or $n-d-1=\sum_{j=1}^r(-1)^{j+1}i_j$, since $G_{2d+1}(\mathbb{R}^{2n})$ and $G_{2d'+1}(\mathbb{R}^{2n}),$ with $d'=n-d-1$, can be identified via $V\mapsto V^\perp$. However, it follows from the same reasoning as in the proof of Theorem \[classesproj\] that, in the second case, $P^{\sigma_\varrho}_{\zeta'}$ does not coincide with the connected component $P_{\zeta_d}^{\sigma_\varrho}$. So we only consider the $\varrho$-semi-canonical elements $\zeta'$ with $d=\sum_{j=1}^r(-1)^{j+1}i_j$.

Set $l=\max I_2$. Next we check that the pair $\zeta'\preceq \tilde\zeta=\sum_{i\in I_1}\zeta_i+\zeta_l$ satisfies the conditions of Proposition \[norm2\]. Considering the same notations we used in the proof of Theorem \[classesproj\], for each $i>0$ we have $$\Delta'_\varrho\cap \Delta_{\zeta'}^{2i}=\{L_j-L_n|\,\,\,   2|I_2\cap\{j,\ldots,l\}|+|I_1\cap\{j,\ldots,n-1\}|=2i\}.$$ In particular, for $i>0$ and $\alpha=L_j-L_n\in \Delta'_\varrho\cap \Delta_{\zeta'}^{2i}$, it is clear that $\alpha(\zeta'-\tilde \zeta)/\mathrm{i}\leq 2i-1,$ and consequently $${\mathfrak{g}}_{2i}^{\zeta'}\cap \mathfrak{m}_\varrho^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i}{\mathfrak{g}}^{\zeta'-\tilde\zeta}_j.$$

For $i>0$, we have the decomposition $${\mathfrak{g}}_{2i-1}^{\zeta'}\cap \mathfrak{k}_\varrho^{\mathbb{C}}=\!\!\! \bigoplus_{{\alpha\in \Delta(\mathfrak{k}_\varrho)\cap \Delta_{\zeta'}^{2i-1}}}\!\!\! {\mathfrak{g}}_\alpha\oplus
   \!\!\!  \bigoplus_{{\alpha\in \Delta'_\varrho\cap \Delta_{\zeta'}^{2i-1}}} \!\!\!({\mathfrak{g}}_\alpha\oplus  {\mathfrak{g}}_{\varrho(\alpha)})\cap \mathfrak{k}^{\mathbb{C}}_\varrho.$$ Given $\alpha\in {\mathfrak{g}}_{2i-1}^{\zeta'}$, since $\alpha(\zeta')/\mathrm{i}$ is odd, we must have $\alpha(\zeta_j)\neq 0$ for some $j\in I_1$. Hence $\alpha(\zeta'-\tilde\zeta)/\mathrm{i}<\alpha(\zeta')/\mathrm{i} $ and we conclude that $${\mathfrak{g}}_{2i-1}^{\zeta'}\cap \mathfrak{k}_\varrho^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i-1}{\mathfrak{g}}^{\zeta'-\zeta}_j.$$

The statement of the theorem follows now from Propositions \[norminf\] and \[norm2\].

Next we will study in detail the case $G_3(\mathbb{R}^6)$. Take as base point of $G_3(\mathbb{R}^6)$ the $3$-dimensional real subspace $V_0\oplus V_1\oplus \overline{V}_1$, where $V_1$ is the one-dimensional isotropic subspace spanned by $\overline{u}_1$. This choice establishes the identification $$G_3(\mathbb{R}^6)\cong SO(6)/S(O(3)\times O(3))$$ and the corresponding involution is $\sigma_{\varrho,1}=\mathrm{Ad}(\exp \pi\zeta_1)\circ \sigma_\varrho$. Following our classification procedure, we also identify $G_3(\mathbb{R}^6)$ with $P_{\zeta_1}^{\sigma_\varrho}$ via the totally geodesic embedding . From Theorem \[cangrass\], we have six classes of harmonic maps into $G_3(\mathbb{R}^6)$: $$\begin{aligned}
 &(\zeta_1,\sigma_\varrho),\,\,\,\,  (\zeta_1+\zeta_2,\sigma_\varrho),\,\,\,\, (\zeta_2, \sigma_{\varrho,1}),\,\,\,\,(\zeta_1, \sigma_{\varrho,2}), \,\,\,\, (\zeta_1+\zeta_2,\sigma_{\varrho,2}),\,\,\,\, (\zeta_2,\mathrm{Ad}(\exp\pi\zeta_2)\circ \sigma_{\varrho,1}).\end{aligned}$$

\[36\] Let $\varphi:S^2\to G_3(\mathbb{R}^6)$ be an harmonic map.

1.  If $\varphi$ is associated to the pair $(\zeta_1,\sigma_\varrho)$ then $\varphi$ is $S^1$-invariant and, up to isometry, is given by $$\label{mixedpair}\varphi=V_0\oplus V\oplus \overline{V},$$ where $V$ is a holomorphic isotropic subbundle of $S^2\times V_0^\perp$ of rank $1$ satisfying $\partial V\subseteq \overline{V}^\perp$.

2.  If $\varphi$ is associated to the pair $(\zeta_1+\zeta_2,\sigma_\varrho)$ and is $S^1$-invariant, then, up to isometry, $$\label{varphi1}
        \varphi=V_0\oplus (W\cap V^\perp) \oplus (\overline {W\cap V^\perp }),$$ where $V\subset W$ are holomorphic isotropic subbundles of $S^2\times V_0^\perp$ of rank $1$ and $2$, respectively, satisfying $\partial V\subset W$ and $\partial W\subset \overline W^\perp.$

3.  If $\varphi$ is associated to the pair $(\zeta_2,\sigma_{\varrho,1})$ and is $S^1$-invariant, then, up to isometry, $$\label{varphi2}\varphi= \{(L_1\oplus \overline{L}_1)^\perp\cap (V_0\oplus V_1\oplus\overline{V}_1)\}\oplus(L_2\oplus \overline{L}_2),$$ where $L_1$ and $L_2$ are holomorphic isotropic bundle lines of $S^2\times (V_0\oplus V_1\oplus\overline{V}_1)$ and $S^2\times (V_0\oplus V_1\oplus\overline{V}_1)^\perp$, respectively.

The corresponding extended solutions have uniton number $2$, $4$, and $2$, respectively, with respect to the standard representation of $SO(6)$. The harmonic maps in the classes $(\zeta_1, \sigma_{\varrho,2})$, $(\zeta_1+\zeta_2,\sigma_{\varrho,2})$, and $(\zeta_2,\mathrm{Ad}(\exp\pi\zeta_2)\circ \sigma_{\varrho,1})$ are precisely the orthogonal complements of the harmonic maps in the classes $(\zeta_1,\sigma_\varrho)$, $(\zeta_1+\zeta_2,\sigma_\varrho)$, and $(\zeta_2, \sigma_{\varrho,1})$, respectively.

For the first two classes, and according to our classification procedure, we identify $G_3(\mathbb{R}^6)$ with $P_{\zeta_1}^{\sigma_\varrho}$ via the totally geodesic embedding $g\cdot (V_0\oplus V_1\oplus \overline{V}_1)\mapsto \exp(\pi\zeta_1)g\sigma_{\varrho,1}(g^{-1}).$ In these two cases, $T_{\sigma_\varrho}$-invariant extended solutions $\Phi$ associated to harmonic maps $\varphi=g\cdot (V_0\oplus V_1\oplus \overline{V}_1)$ satisfy $$\label{phimenos1}
\Phi_{-1}=\exp(\pi\zeta_1)g\sigma_{\varrho,1}(g^{-1})=\exp(\pi\zeta_1)(\pi_\varphi-\pi_{\varphi}^\perp)\exp(\pi\zeta_1)s_0.$$

First we consider the harmonic maps associated to the pair $(\zeta_1,\sigma_\varrho)$. We have $r(\zeta_1)=1$ and $$(\mathfrak{u}^0_{\zeta_1})_{\sigma_{\varrho}}=(\mathfrak{p}^{\zeta_1}_{0})^\perp\cap \mathfrak{k}^{\mathbb{C}}_{{\varrho}}={\mathfrak{g}}_1^{\zeta_1}\cap \mathfrak{k}^{\mathbb{C}}_{{\varrho}}.$$ Consequently any such harmonic map is $S^1$-invariant. Write $\gamma_{\zeta_1}(\lambda)=\lambda^{-1}\pi_{V_1}+\pi^\perp_{V_1\oplus \overline{V}_1}+\lambda \pi_{\overline{V}_1},$ where $V_1$ is the one-dimensional isotropic space spanned by $\overline{u}_1$. Let $\Phi:S^2\setminus D\to U_{\zeta_1}^{\sigma_\varrho}$ be an extended solution associated to the harmonic map $\varphi$. Then, by $S^1$-invariance, we can write $$\label{phicase1}
 \Phi_\lambda=\lambda^{-1}\pi_V+ \pi^\perp_{V\oplus \overline{V}}+\lambda\pi_{\overline V},$$ where $V$ is a holomorphic isotropic subbundle of $S^2\times \mathbb{R}^{6}$ of rank $1$ satisfying $\partial V\subseteq \overline{V}^\perp$. The $T_{\sigma_\varrho}$-invariance of $\Phi$ implies that $V_0\subset (V\oplus\overline{V})^\perp$. Equating and , we get, up to isometry, $\varphi=V_0\oplus V\oplus \overline{V}$.

For the case $(\zeta_1+\zeta_2,\sigma_{\varrho})$, since $$\label{zeta1+zeta2}
\gamma_{\zeta_1+\zeta_2}(\lambda)=\lambda^{-2}\pi_{V_1}+\lambda^{-1}\pi_{V_2\cap V_1^\perp}+\pi_{V_2\oplus \overline{V}_2}^\perp+\lambda\pi_{\overline{V}_2\cap \overline{V}_1^\perp}+\lambda^2\pi_{\overline{V}_1},$$ any $S^1$-invariant harmonic map $\varphi$ in this class admits an extended solution of the form $$\label{phicase2}
\Phi_\lambda= \lambda^{-2}\pi_{V}+\lambda^{-1}\pi_{W\cap V^\perp}+\pi_{W\oplus \overline{W}}^\perp+\lambda\pi_{\overline{W}\cap \overline{V}^\perp}+\lambda^2\pi_{\overline{V}},$$ where $V\subset W$ are holomorphic isotropic subbundles of rank $1$ and $2$, respectively, satisfying $\partial V\subset W$ and $\partial W\subset \overline W^\perp.$ By $T_{\sigma_\varrho}$-invariance, we must have $V_0\subset (W\oplus\overline{W})^\perp$, hence $V\subset W$ are subbundles of $S^2\times V_0^\perp$. Equating and , we get .

For the case $(\zeta_2,\sigma_{\varrho,1})$, we identify $G_3(\mathbb{R}^6)$ with $P_{\zeta_2}^{\sigma_{\varrho,1}}=\exp\pi\zeta_1P_{\zeta_2-\zeta_1}^{\sigma_\varrho}$ via the totally geodesic embedding $$\label{oao}
g\cdot (V_0\oplus V_1\oplus \overline{V}_1)\mapsto g\sigma_{\varrho,1}(g^{-1}).$$ Extended solutions $\Phi$ associated to $S^1$-invariant harmonic maps in this class must be of the form $$\label{phiws}
\Phi_\lambda=\lambda^{-1}\pi_W+ \pi_{W\oplus \overline W}^\perp+\lambda\pi_W,$$ where $W$ is a holomorphic isotropic subbundle of rank $2$. By $T_{\sigma_{\varrho,1}}$-invariance, we must have $[\pi_W,\pi_{V_0\oplus  V_1\oplus\overline{V}_1}]=0,$ which means that $W$ must be of the form $W=L_1\oplus L_2$, where $L_1$ and $L_2$, respectively, are holomorphic isotropic bundle lines of $S^2\times (V_0\oplus V_1\oplus\overline{V}_1)$ and $S^2\times (V_0\oplus V_1\oplus\overline{V}_1)^\perp$.

On the other hand, in view of , we have $\Phi_{-1}=(\pi_\varphi-\pi_\varphi^\perp)\exp(\pi\zeta_1) s_0.$ Equating this with , we conclude that holds. The remaining cases are treated similarly.

The first two classes of $S^1$-invariant harmonic maps $\varphi:S^2\to G_3(\mathbb{R}^6)$ in Theorem \[36\] factor through $G_2(\mathbb{R}^5)$. That is, for any such harmonic map $\varphi$, there exists $\tilde{\varphi}:S^2\to G_2(\mathbb{R}^5)$, where we identify $\mathbb{R}^5$ with $V_0^\perp$, such that $\varphi=V_0\oplus \tilde{\varphi}$. An explicit construction of all harmonic maps from $S^2$ into $G_2(\mathbb{R}^n)$ can be found in [@woodG2]. In that paper, harmonic maps of the form are called *real mixed pairs*. We emphasise that the harmonic maps into $G_3(\mathbb{R}^6)$ associated to extended solutions in the corresponding unstable manifolds need not to factor through $G_2(\mathbb{R}^5)$ in the same way.

Let us consider the case $(\zeta_1+\zeta_2,\sigma_{\varrho})$. Taking into account the Weierstrass representation of Theorem \[sigmaweirstrass\], any extended solution $\Phi:S^2\setminus D\to U^{\sigma_{\varrho}}_{\zeta}(SO(6))$, with $\zeta=\zeta_1+\zeta_2$, can be written as $\Phi=\exp{C}\cdot \gamma_{\zeta}$, for some meromorphic vector-valued function $C:S^2\to (\mathfrak{u}^0_\zeta)_{\sigma_\varrho}$. We have $r(\zeta)=3$ and $$(\mathfrak{u}^0_{\zeta})_{\sigma_\varrho}=({\mathfrak{g}}_1^\zeta\oplus{\mathfrak{g}}_2^\zeta\oplus {\mathfrak{g}}_3^\zeta)\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}\oplus \lambda ({\mathfrak{g}}_2^\zeta\oplus {\mathfrak{g}}_3^\zeta)\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}\oplus \lambda^2 {\mathfrak{g}}_3^\zeta\cap  \mathfrak{k}^{\mathbb{C}}_{\varrho}.$$ Moreover, $$\begin{aligned}
&{\mathfrak{g}}_1^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}={\mathfrak{g}}_{L_1-L_2}\oplus \{({\mathfrak{g}}_{L_2- L_3}\oplus{\mathfrak{g}}_{L_2+L_3}) \cap \mathfrak{k}^{\mathbb{C}}_{\varrho}\},\quad
{\mathfrak{g}}_2^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}=({\mathfrak{g}}_{L_1+ L_3}\oplus{\mathfrak{g}}_{L_1-L_3} )\cap \mathfrak{k}^{\mathbb{C}}_{\varrho},\\
& {\mathfrak{g}}_3^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}={\mathfrak{g}}_{L_1+L_2},\quad
({\mathfrak{g}}_2^\zeta\oplus {\mathfrak{g}}_3^\zeta)\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}={\mathfrak{g}}_2^\zeta\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}=({\mathfrak{g}}_{L_1- L_3}\oplus {\mathfrak{g}}_{L_1+ L_3})\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}.\end{aligned}$$

Write $$\label{CC}
C=C_0+\lambda C_1+\lambda^2 C_2,\quad C_0=c_0^1+c_0^2+c_0^3,\quad C_1=c_1^2+c_1^3,\quad C_2=c_2^3$$ where the functions $c_0^i:S^2\to {\mathfrak{g}}_i^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}$, $c_1^i:S^2\to {\mathfrak{g}}_i^\zeta\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}$, and $c_2^3:S^2\to {\mathfrak{g}}_3^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}$ are meromorphic functions. Clearly, $c_1^3=0$. Consider the root vectors defined by . Since $\sigma_\varrho(X_{2,3})=-Y_{2,3}$ and $\sigma_\varrho(X_{1,3})=-Y_{1,3}$, we can write $$\begin{aligned}
  c_0^1=aX_{1,2}+b(X_{2,3}-Y_{2,3}),\,\,c_0^2=c(X_{1,3}-Y_{1,3}),\,\, c_0^3=dY_{1,2},\,\,
  c_1^2=e(X_{1,3}+Y_{1,3}),\,\,c_2^3=fX_{1,2}\end{aligned}$$ in terms of ${\mathbb{C}}$-valued meromorphic functions $a$, $b$, $c$, $d$, $e$, $f$.

Taking into account the results of Section \[weircond\], $\Phi=\exp{C}\cdot \gamma_{\zeta}$ is an extended solution if and only if, in the expression $$(\exp C)^{-1}(\exp C)_z=C_z-\frac{1}{2!}(\mathrm{ad} C)C_z+\frac{1}{3!}(\mathrm{ad} C)^2C_z,$$ we have:

1.  the independent coefficient should have zero component in each ${\mathfrak{g}}_{2}^{\zeta}$ and ${\mathfrak{g}}_{3}^{\zeta},$ that is $$\begin{aligned}
    \label{o}
        {c_0^2}_z-\frac{1}{2}[c_0^1,{c_0^1}_z]=0,\,\,\,\, {c_0^3}_z-\frac{1}{2}[c_0^1,{c_0^2}_z]-\frac{1}{2}[c_0^2,{c_0^1}_z]+\frac16[c_0^1,[c_0^1,{c_0^1}_z]]=0;
      \end{aligned}$$

2.  the $\lambda$ coefficient should have zero component in ${\mathfrak{g}}_{3}^{\zeta},$ that is $$\label{1}
         [c_0^1,{c_1^2}_z]+[c_1^2,{c_0^1}_z]=0.$$

From equations we get the equations (prime denotes $z$-derivative) $$\label{merocond}
2c'=ab'-ba',\,\,\,\,\, 3d'=3 cb'-bc';$$ on the other hand, observe that always holds since $$[c_0^1,{c_1^2}_z]+[c_1^2,{c_0^1}_z]\in [{\mathfrak{g}}_1^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho},{\mathfrak{g}}_2^\zeta\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}]\subset {\mathfrak{g}}_3^\zeta\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}=\{0\}.$$

Hence we conclude that, any extended solution $\Phi:S^2\setminus D\to U^{\sigma_{\varrho}}_{\zeta}(SO(6))$, with $\zeta=\zeta_1+\zeta_2$, of the form $\Phi=\exp{C}\cdot \gamma_{\zeta}$, can be constructed as follows: choose arbitrary meromorphic functions $a$, $b$, $e$ and $f$; integrate equations to obtain the meromorphic functions $c$ and $d$; $C$ is then given by .

Choose $a(z)=b(z)=z$. From , we can take $c(z)=1$ and $d(z)=z$. This data defines the matrix $C_0$ and the $S^1$-invariant extended solution $\Phi^0=\exp C_0\cdot \gamma_{\zeta}$, where the loop $\gamma_\zeta$, with $\zeta=\zeta_1+\zeta_2$, is given by . The extended solutions $\Phi:S^2\to U_{\zeta}^{\sigma_\varrho}(SO(6))$ satisfying $\Phi^0=u_{\zeta}\circ\Phi$ are of the form $\Phi=\exp C\cdot \gamma_\zeta$, where the matrix $C=C_0+C_1\lambda+C_2\lambda^2$ is given by $$C= \left(
          \begin{array}{cccccc}
            0 & z & 1 & 0 & z & -1 \\
            0 & 0 & z & -z & 0 & -z \\
            0 & 0 & 0 & 1 & z & 0 \\
            0 & 0 & 0 & 0 & 0 & 0 \\
            0 & 0 & 0 & -z & 0 & 0 \\
            0 & 0 & 0 & -1 & -z & 0 \\
          \end{array}
        \right)+ \left(
          \begin{array}{cccccc}
            0 & 0 & e\lambda & 0 & f\lambda^2 & -e\lambda \\
            0 & 0 & 0 & -f\lambda^2 & 0 & 0 \\
            0 & 0 & 0 & e\lambda & 0 & 0 \\
            0 & 0 & 0 & 0 & 0 & 0 \\
            0 & 0 & 0 & 0 & 0 & 0 \\
            0 & 0 & 0 & -e\lambda & 0 & 0 \\
          \end{array}
        \right),$$ with respect to the complex orthonormal basis $\mathbf{u}=\{u_1,u_2,u_3,\overline{u}_1,\overline{u}_2,\overline{u}_3\}$, where $e$ and $f$ are arbitrary meromorphic functions on $S^2$. The holomorphic vector bundles $V$ and $W$ associated to the $S^1$-invariant extended solution $\exp C_0\cdot \gamma_{\zeta}$ are given by $V=\exp{C_0}\cdot {V}_1$ and $W=\exp{C_0}\cdot V_2$, and we have, with respect to the basis $\mathbf{u}$, $$\begin{aligned}
  V&=\mathrm{span}\{(12-12z^2-z^4,-4z^3,12-6z^2,12,-12z,-12+6z^2 ) \}\\
  W&=\mathrm{span}\{(6z+z^3,3z^2,3z,0,3,-3z ) \}\oplus V.\end{aligned}$$

Outer symmetric $SU(2n+1)$-spaces. {#su(2n+1)-outer}
----------------------------------

Let $E_j$ be the square $(m\times m)$-matrix whose $(j,j)$-entry is $\mathrm{i}$ and all other entries are $0$. The complexification ${\mathfrak{t}}^{\mathbb{C}}$ of the algebra ${\mathfrak{t}}$ of diagonal matrices $\sum a_iE_i$, with $a_i\in \mathbb{C}$ and $\sum a_i=0$, is a Cartan subalgebra of $\mathfrak{su}(m)^{\mathbb{C}}$. Let $\{L_1,\ldots,L_{m}\}$ be the dual basis of $\{E_1,\ldots,E_{m}\}$, that is $L_i(E_j)=\mathrm{i}\delta_{ij}$. The roots of $\mathfrak{su}(m)$ are the vectors $ L_i- L_j$, with $i\neq j$ and $1\leq i,j\leq m-1$ and $\Delta^+=\{L_i-L_j\}_{i<j}$ is a positive root system with positive simple roots $\alpha_i=L_i-L_{i+1}$, for $1\leq i\leq m-1$. For $i\neq j$, the matrix $X_{i,j}$ whose $(i,j)$ entry is $1$ and all other entries are $0$ generate the root space ${\mathfrak{g}}_{L_i-L_j}$. The dual basis of $\Delta_0=\{\alpha_1,\ldots,\alpha_{m-1}\}$ in $\mathrm{i}\mathfrak{t}^*$ is formed by the matrices $$\begin{aligned}
H_i=\frac{m-i}{m}(E_1+\ldots+ E_i)- \frac{i}{m}(E_{i+1}+\ldots+ E_{m}).\end{aligned}$$

### Special Lagrangian spaces

Consider on $\mathbb{R}^{2m}$ the standard inner product $\langle\cdot,\cdot\rangle$ and the canonical orthonormal basis $\mathbf{e}^{2m}=\{e_1,\ldots, e_{2m}\}.$ Define the orthogonal complex structure $I$ by $I(e_i)=e_{2m+1-i}$, for $i\in \{1,\ldots,m\}$. A *Lagrangian subspace* of $\mathbb{R}^{2m}$ (with respect to $I$) is a $m$-dimensional subspace $L$ such that $IL\perp L$. Let $\mathcal{L}_m$ be the space of all Lagrangian subspaces of $\mathbb{R}^{2m}$ and $L_0\in\mathcal{L}_m$ the Lagrangian subspace generated by $\mathbf{e}^{m}=\{e_1,\ldots,e_m\}$. The unitary group $U(m)$ acts transitively on $\mathcal{L}_m$, with isotropy group at $L_0$ equal to $SO(m)$, and $\mathcal{L}_m$ is a reducible symmetric space that can be identified with $U(m)/SO(m)$ (see [@Zi] for details).

The space $\mathcal{L}_m$ can also be interpreted as the set of all orthogonal linear maps $\tau:\mathbb{R}^{2m}\to\mathbb{R}^{2m}$ satisfying $\tau^2=e$ and $I\tau=-\tau I$. Indeed, if $V_\pm$ are the $\pm 1$ eigenspaces of $\tau$, then $IV_+=V_-$ and $IV_+\perp V_+$, that is $V_+$ is Lagrangian. From this point of view, $U(m)$ acts on $\mathcal{L}_m$ by conjugation: $g\cdot \tau=g\tau g^{-1}$. Let $\tau_0\in \mathcal{L}_m$ be the orthogonal linear map corresponding to $L_0$, that is, ${\tau_0}_{|_{L_0}}=e$ and ${\tau_0}_{|_{IL_0}}=-e$. The corresponding involution on $U(m)$ is given by $\sigma(g)=\tau_0g\tau_0$ and the Cartan embedding $\iota:\mathcal{L}_m\hookrightarrow U(m)$ is given by $\iota(\tau)=\tau \tau_0$.

The totally geodesic submanifold $\mathcal{L}^s_m:=SU(m)/SO(m)$ of $U(m)/SO(m)$ is also known as the *space of special Lagrangian subspaces of $\mathbb{R}^{2m}$*. It is an irreducible outer symmetric $SU(m)$-space.

### Harmonic maps into $\mathcal{L}^s_{2n+1}$

Take $m=2n+1$. The non-trivial involution $\varrho$ of the Dynkin diagram of $\mathfrak{su}(2n+1)^{\mathbb{C}}$ is given by $\varrho(\alpha_i)=\alpha_{2n+1-i}$. In particular, $\varrho$ does not fix any root in $\Delta_0$ and there exists only one class of outer symmetric $SU(2n+1)$-spaces. The semi-fundamental basis $\pi_{\mathfrak{k}_\varrho}(\Delta_0)=\{\beta_1,\ldots,\beta_{n}\}$ is given by $\beta_i=\frac12(\alpha_i+\alpha_{2n+1-i})$ whereas the dual basis $\{\zeta_1,\ldots,\zeta_{n}\}$ is given by $$\zeta_i=H_i+H_{2n+1-i}=E_1+\ldots+E_i-(E_{2n+2-i}+\ldots +E_{2n+1}),$$ for $1\leq i\leq n$. Since each $\zeta_i$ belongs to the integer lattice $\mathfrak{I}(SU(2n+1))$, the $\varrho$-semi-canonical elements of $SU(2n+1)$ are precisely the elements $\zeta=\sum_{i=1}^{n}m_i\zeta_i$ with $m_i\in\{0,1,2\}$.

Let $\mathbf{e}^{2n+1}=\{e_1,\ldots,e_{2n+1}\}$ be the canonical orthonormal basis of $\mathbb{R}^{2n+1}$. Identify ${\mathbb{C}}^{2n+1}$ with $(\mathbb{R}^{4n+2},I)$, where $I$ is defined as above. Set $$v_j=\frac{1}{\sqrt2}(e_j+\mathrm{i}e_{2n+2-j}),$$ for $1\leq j\leq n$, $v_{n+1}=e_{n+1}$ and $v_{2n+2-j}=\overline{v}_j$. Take the matrices $E_j$ with respect to the complex basis $\mathbf{v}=\{v_1,\ldots,v_{2n+1}\}$ of ${\mathbb{C}}^{2n+1}$. Hence $\tau_0E_j\tau_0=-E_{2n+2-j}$ and the fundamental involution $\sigma_\varrho$ is given by $\sigma_\varrho(g)=\tau_0g\tau_0$. The fundamental outer symmetric $SU(2n+1)$-space is the space of special Lagrangian subspaces $\mathcal{L}_{2n+1}^s=SU(2n+1)/SO(2n+1)$, and this is the unique outer symmetric $SU(2n+1)$-space.

Next we consider in detail harmonic maps into $\mathcal{L}_{3}^s$. In this case we have two non-zero $\varrho$-semi-canonical elements, $\zeta_1$ and $2\zeta_1$, and consequently two classes of harmonic maps, $(\zeta_1,\sigma_\varrho)$ and $(\zeta_1,\sigma_{\varrho,1})$. Since $\zeta_1=E_1-E_3$, we have $r(\zeta_1)=(L_1-L_3)(\zeta_1)/\mathrm{i}=2$. Let $W_1$, $W_2$ and $W_3$ be the complex one-dimensional images of $E_1$, $E_2$ and $E_3$, respectively. Any extended solution $$\Phi:S^2\setminus D\to U_{\zeta_1}^{\sigma_\varrho}(SU(2n+1))$$ is given by $\Phi=\exp C\cdot \gamma_{\zeta_1}$, with $\gamma_{\zeta_1}(\lambda)=\lambda^{-1}\pi_{W_3}+\pi_{W_2}+\lambda \pi_{W_1}$, for some holomorphic vector-valued function $C:S^2\setminus D\to (\mathfrak{u}^0_{\zeta_1})_{\sigma_\varrho}$, where $$(\mathfrak{u}^0_{\zeta_1})_{\sigma_\varrho}=(\mathfrak{p}_0^{\zeta_1})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\varrho}+\lambda(\mathfrak{p}_1^{\zeta_1})^\perp\cap
\mathfrak{m}^{\mathbb{C}}_{\varrho}$$ and $$\begin{aligned}
 (\mathfrak{p}_0^{\zeta_1})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\varrho}=({\mathfrak{g}}_{L_1-L_2}\oplus{\mathfrak{g}}_{L_2-L_3}\oplus {\mathfrak{g}}_{L_1-L_3})\cap\mathfrak{k}^{\mathbb{C}}_{\varrho},\quad (\mathfrak{p}_1^{\zeta_1})^\perp\cap\mathfrak{m}^{\mathbb{C}}_{\varrho}= {\mathfrak{g}}_{L_1-L_3}\cap\mathfrak{m}^{\mathbb{C}}_{\varrho}.\end{aligned}$$

Let $X_{i,j}$ be the square matrix whose $(i,j)$ entry is $1$ and all the other entries are $0$, with respect to the basis $\mathbf{v}$. The root space $\mathfrak{g}_{L_i-L_j}$ is spanned by $X_{i,j}$. We have $\sigma_{\varrho}(X_{1,2})=-X_{2,3}$ and $\sigma_\varrho(X_{1,3})= -X_{1,3}$ (consequently, $\mathfrak{g}_{L_1-L_3}\subset \mathfrak{m}_{\varrho}^{\mathbb{C}}$). Hence we can write $C=C_0+C_1\lambda$, with $C_0=a(X_{1,2}-X_{2,3})$ and $C_1=bX_{1,3}$, for some meromorphic functions $a,b$ on $S^2$. The harmonicity equations do not impose any condition on these meromorphic functions, hence any harmonic map $\varphi:S^2\to \mathcal{L}_{3}^s$ in the class $(\zeta_1,\sigma_\varrho)$ admits an extended solution of the form $$\label{su1}
\Phi=\exp\left(
            \begin{array}{ccc}
              0 & a & b\lambda \\
              0 & 0 & -a \\
              0 & 0 & 0 \\
            \end{array}
          \right)\cdot \gamma_{\zeta_1}=\left(
                                          \begin{array}{ccc}
                                            1 & a & \frac12(-a^2+2b\lambda) \\
                                            0 & 1 & -a \\
                                            0 & 0 & 1 \\
                                          \end{array}
                                        \right)\cdot \gamma_{\zeta_1},$$ and $\varphi$ is recovered by setting $\varphi=\Phi_{-1}\tau_0$. Similarly, one can see that the class of harmonic maps in $(\zeta_1,\sigma_{\varrho,1})$ admits an extended solution of the form $$\label{su2}\Phi=\left(
                                          \begin{array}{ccc}
                                            1 & a & \frac12(a^2+2b\lambda) \\
                                            0 & 1 & a \\
                                            0 & 0 & 1 \\
                                          \end{array}
                                        \right)\cdot \gamma_{\zeta_1},$$ with no restrictions on the meromorphic functions $a$ and $b$.

H. Ma established (cf. Theorem 4.1 of [@Ma]) that harmonic maps $\varphi:S^2\to \mathcal{L}_{3}^s$ are essentially of two types: 1) $\iota_\sigma\circ \varphi$ is a *Grassmannian solution* obtained from a full harmonic map $f:S^2\to \mathbb{R}P^2\subset \mathbb{C}P^2$, where $\iota_\sigma$ is the Cartan embedding of $ \mathcal{L}_{3}^s$ in $SU(3)$; 2) up to left multiplication by a constant, $\iota_\sigma\circ \varphi$ is of the form $(\pi_{\beta_1}- \pi_{\beta_1}^\perp) (\pi_{\beta_2}- \pi_{\beta_2}^\perp)$, where $\beta_1$ is a *Frenet pair* associated to a full totally istotropic holomorphic map $g:S^2\to\mathbb{C}P^2$ and $\beta_2$ is a rank $1$ holomorphic subbundle of $G'(g)^\perp$, where $G'(g)$ is the first *Gauss bundle* of $g$. Observe that if, in the second case, $\beta_2$ coincides with $g$, then $\iota_\sigma\circ\varphi$ is a [Grassmannian solution]{} obtained from the full harmonic map $f:=G'(g)$ from $S^2$ to $\mathbb{R}P^2$, that is, $\varphi$ is of type 1). Comparing this with our description, it is not difficult to see that harmonic maps of type 1) are $S^1$-invariant extended solutions (take $b=0$ in and ) and harmonic maps of type 2) are associated to extended solutions with values in the corresponding unstable manifolds (which corresponds to an arbitrary choice of $b$ in and ). H. Ma also established a purely algebraic explicit construction of such harmonic maps in terms of meromorphic data on $S^2$, which is consistent with our results.

Outer symmetric $SU(2n)$-spaces.
--------------------------------

With the same notations of Section \[su(2n+1)-outer\], the non-trivial involution $\varrho$ of the Dynkin diagram of $\mathfrak{su}(2n)$ is given by $\varrho(\alpha_i)=\alpha_{2n-i}$, and $\varrho$ fixes the root $\alpha_n$. The semi-fundamental basis $\pi_{\mathfrak{k}_\varrho}(\Delta_0)=\{\beta_1,\ldots,\beta_{n-1}\}$ is given by $\beta_1=\alpha_n$ and $\beta_i=\frac12(\alpha_i+\alpha_{2n-i})$ if $i\geq 2$; whereas its dual basis $\{\zeta_1,\ldots,\zeta_{n-1}\}$ is given by $$\begin{aligned}
  \zeta_1&=H_n=\frac12({E_1}+\ldots+E_n)-\frac12(E_{n+1}+\ldots+ E_{2n})\\
  \zeta_i&= H_{i-1}+H_{2n-i+1}=E_1+\ldots+E_{i-1}-(E_{2n+2-i}+\ldots+E_{2n}),\,\,\,\,\mbox{for $2\leq i\leq n-1$}.\end{aligned}$$

By Theorem \[murak\], there exist two conjugacy classes of outer involutions: the fundamental outer involution $\sigma_\varrho$ and $\sigma_{\varrho,1}$. These outer involutions correspond to the symmetric spaces $SU(2n)/Sp(n)$ and $SU(2n)/SO(2n)$, respectively. Observe that $\zeta_1$ does not belong to the integer lattice $\mathfrak{I}'(SU(2n)^{\sigma_\varrho})$ since $\exp 2\pi\zeta_1=-e$.

### Harmonic maps into the space of special unitary quaternionic structures on $\mathbb{C}^{2n}$. 

A *unitary quaterninonic structure* on the standard hermitian space $(\mathbb{C}^{2n},\langle\cdot,\cdot\rangle )$ is a conjugate linear map $J:{\mathbb{C}}^{2n}\to {\mathbb{C}}^{2n}$ satisfying $J^2=-Id$ and $\langle v,w\rangle= \langle J\,w,J\,v\rangle$ for all $v,w\in {\mathbb{C}}^{2n}$. Consider as base point the quaternionic structure $J_o$ defined by $J_o(e_i)=e_{2n+1-i}$ for each $1\leq i\leq n$, where $\mathbf{e}^{2n}=\{e_1,\ldots,e_{2n}\}$ is the canonical hermitian basis of $\mathbb{C}^{2n}$. The unitary group $U(2n)$ acts transitively on the space of unitary quaternionic structures on ${\mathbb{C}}^{2n}$ with isotropy group at $J_o$ equal to $Sp(n)$, and thus $M=U(2n)/Sp(n)$. This is a reducible symmetric space with involution $\sigma:U(2n)\to U(2n)$ given by $\sigma(X)=J_oXJ_o^{-1}$, but the totally geodesic submanifold $\mathcal{Q}^s_n:=SU(2n)/Sp(n)$ is an irreducible symmetric space, which we call the space of *special unitary quaternionic structures* on $\mathbb{C}^{2n}$ (see [@Zi] for details). If we consider the matrices $E_i$ with respect to the complex basis $\mathbf{v}=\{v_1,\ldots,v_{2n}\}$ defined by $$\label{ves}
 v_j=\frac{1}{\sqrt2}(e_j+\mathrm{i}e_{2n+1-j}),$$ for $1\leq j\leq n$, and $v_{2n+1-j}=\overline{v}_j$, we see that $J_oE_jJ_o^{-1}=-E_{2n+1-j}$, and consequently we have $\sigma=\sigma_\varrho$.

Next we consider with detail harmonic maps into $\mathcal{Q}^s_2$.

Each harmonic map $\varphi:S^2\to\mathcal{Q}^s_2$ belongs to one of the following classes: $(2\zeta_1,\sigma_\varrho)$, and $(\zeta_2,\sigma_{\varrho,2})$.

We start by identifying $\mathcal{Q}^s_2$ with $P_e^{\sigma_\varrho}$.

The $\varrho$-semi-canonical elements of $SU(4)$ are precisely the elements $$2\zeta_1,\,4\zeta_1,\,\zeta_2,\,2\zeta_2,\,2\zeta_1+\zeta_2,\,2\zeta_1+2\zeta_2,\,4\zeta_1+\zeta_2,\,4\zeta_1+2\zeta_2.$$ By Proposition \[fundcan\], all these elements correspond to the symmetric space $\mathcal{Q}^s_2$.

We claim that $\exp\pi \zeta_2$ is not in the connected component $$P_e^{\sigma_\varrho}=\{gJ_og^{-1}J_o^{-1}|\,g\in SU(4)\}.$$ In fact, $\exp(\pi \zeta_2)J_o=gJ_og^{-1}\cong gSp(n)$ for the unitary transformation $g$ defined by $g(e_1)=e_4$, $g(e_4)=e_1$, $g(e_2)=e_3$ and $g(e_3)=-e_2$. Since $\det g\neq 1$ we conclude that $\exp\pi \zeta_2$ does not belong to $P_e^{\sigma_\varrho}$. Similarly, one can check that $\exp\pi (2\zeta_1+\zeta_2)$ is not in $P_e^{\sigma_\varrho}$.

Hence, since $\exp\pi2\zeta_1$ belongs to the centre of $SU(4)$, any harmonic map $\varphi:S^2\to\mathcal{Q}^s_2\cong P_e^{\sigma_\varrho} $ belongs to one of the following classes: $(2\zeta_1,\sigma_\varrho)$, $(\zeta_2,\sigma_{\varrho,2})$, and $(2\zeta_1+\zeta_2,\sigma_{\varrho,2})$. It remains to check that, in view of Proposition \[norm2\], harmonic maps in the class $(2\zeta_1+\zeta_2,\sigma_{\varrho,2})$ can be normalized to harmonic maps in the class $(\zeta_2,\sigma_{\varrho,2})$.

It is clear that $2\zeta_1+\zeta_2\preceq \zeta_2$. On the other hand, for any positive root $L_i-L_j\in \Delta^+$, with $i<j$, we have $(L_i-L_j)(2\zeta_1)/\mathrm{i}\leq (L_i-L_j)({2\zeta_1+\zeta_2})/\mathrm{i},$ where the equality holds in just one case: $(L_2-L_3)(2\zeta_1)= (L_2-L_3)(2\zeta_1+\zeta_2)=2\mathrm{i}.$ However, $\mathfrak{g}_{L_2-L_3}\subset \mathfrak{k}_{\sigma_{\varrho,2}}$, which means that the conditions of Proposition \[norm2\] hold for $\zeta=2\zeta_1+\zeta_2$ and $\zeta'=\zeta_2$, and consequently harmonic maps in the class $(2\zeta_1+\zeta_2,\sigma_{\varrho,2})$ can be normalized to harmonic maps in the class $(\zeta_2,\sigma_{\varrho,2})$.

Following the same procedure as before, one can see that any harmonic map $\varphi\to\mathcal{Q}^s_2$ in the class $(2\zeta_1,\sigma_\varrho)$ admits an extended solution of the form $$\Phi=\left(
          \begin{array}{cccc}
            1 & 0 & c_1+a\lambda & c_2 \\
            0 & 1 & c_3 & c_1-a\lambda \\
            0 & 0 & 1 & 0 \\
            0 & 0 & 0 & 1 \\
          \end{array}
        \right)\cdot \gamma_{2\zeta_1},$$ where $c_1,c_2,c_3\in{\mathbb{C}}$ are constants, $a$ is a meromorphic function on $S^2$. The harmonic map is recovered by setting $\varphi=\Phi_{-1}J_o$. Reciprocally, given arbitrary complex constants $c_1,c_2,c_3$ and a meromorphic function $a:S^2\to {\mathbb{C}}$, such $\Phi$ is an extended solution associated to some harmonic map in the class $(2\zeta_1,\sigma_\varrho)$ (the harmonicity equations do not impose any restriction to $a$).

Similarly, any harmonic map $\varphi\to\mathcal{Q}^s_2$ in the class $(\zeta_2,\sigma_{\varrho,2})$ admits an extended solution of the form $$\Phi=\left(
          \begin{array}{cccc}
            1 & b & a & c \\
            0 & 1 & 0 & a \\
            0 & 0 & 1 & -b \\
            0 & 0 & 0 & 1 \\
          \end{array}
        \right)\cdot \gamma_{\zeta_2},$$ where $a$, $b$ and $c$ are meromorphic functions satisfying $c'=ba'-b'a$. Since $P_{\zeta_2}^{\sigma_{\varrho,2}}=\exp(\pi\zeta_2)P_e^{\sigma_\varrho}$, the harmonic map is recovered by setting $\varphi=\exp(\pi\zeta_2)\Phi_{-1}J_o$.

### Harmonic maps into $\mathcal{L}^s_{2n}$. 

The outer symmetric $SU(2n)$-space that corresponds to the involution $\sigma_{\varrho,1}$ is the space of special Lagrangian subspaces $\mathcal{L}^s_{2n}\cong SU(2n)/SO(2n)$. Take as base point the Lagrangian space $L_o=\mathrm{Span}\{e_1,\ldots,e_{2n}\}$ of $\mathbb{R}^{4n}$ and let $\tau_0$ be the corresponding conjugation, so that the Cartan embedding of $\mathcal{L}^s_{2n}$ into $SU(2n)$ is given by $\tau=g\tau_og^{-1}\mapsto g\tau_0 g^{-1} \tau\in P_e^{\sigma_{\varrho,1}}$.

\[lemafixe\] For each $\zeta\in \mathfrak{I}(SU(2n)^{\sigma_{\varrho,1}})$ we have $\exp\pi\zeta\in  P_e^{\sigma_{\varrho,1}}$.

Each $\zeta\in \mathfrak{I}(SU(2n)^{\sigma_{\varrho,1}})$ can be written as $\zeta=\sum_{i=1}^{n}n_i(E_i-E_{2n+1-i}).$ Hence, $\exp\pi\zeta=\pi_V-\pi_{V}^\perp$, where $V=\bigoplus_{{n_i\, \mathrm{even}}}\mathrm{Span}\{e_i,e_{2n+1-i}\}$. Define $g\in SU(2n)$ as follows: if $n_i$ is even, then $g(e_i)=e_i$ and $g(e_{2n+1-i})=e_{2n+1-i}$; if $n_i$ is odd, then $g(e_i)=\mathrm{i} e_{i}$ and $g(e_{2n+1-i})=-\mathrm{i}e_{2n+1-i}$. We have $\exp\pi\zeta=g\tau_0g^{-1}\tau_0$, that is $\exp\pi\zeta\in P_e^{\sigma_{\varrho,1}}$.

Now, identify $\mathcal{L}^s_{2n}$ with $P_e^{\sigma_{\varrho,1}}$ via its Cartan embedding. By Theorem \[tinva\], any harmonic map $\varphi:S^2\to P_e^{\sigma_{\varrho,1}}$ admits an extended solution $\Phi:S^2\setminus D\to U_{\zeta'}^{\sigma_{\varrho,1}}(SU(2n))$, for some $\zeta'\in \mathfrak{I}'(SU(2n))\cap\mathfrak{k}_{\sigma_{\varrho,1}}$ and some discrete subset $D$. We can assume that $\zeta'$ is a $\varrho$-semi-canonical element. The corresponding $S^1$-invariant solution $u_\zeta\circ \Phi$ takes values in $\Omega_{\xi}(SU(2n)^{\sigma_{\varrho,1}})$, with $\xi\in \mathfrak{I}'(SU(2n)^{\sigma_{\varrho,1}})$; and both $\Phi_{-1}$ and $(u_\zeta\circ \Phi)_{-1}$ take values in $P_\xi^{\sigma_{\varrho,1}}$. A priori, $\xi$ can be different from $\zeta$ since $\sigma_{\varrho,1}$ is not a fundamental outer involution. However, by Lemma \[lemafixe\] we have $P_\xi^{\sigma_{\varrho,1}}=P_e^{\sigma_{\varrho,1}}=P_{\zeta'}^{\sigma_{\varrho,1}}.$

If $\zeta$ is a $\varrho$-canonical element such that $\zeta'\preceq \zeta$ and $\mathcal{U}_{\zeta',\zeta'-\zeta}(\Phi)$ is constant, then, taking into account Proposition \[norminf\], there exists a $T_\tau$-invariant extended solution $\tilde\Phi: S^2\setminus D\to U^\tau_{\zeta}(SU(2n)),$ where $$\label{invo1}\tau=\mathrm{Ad}(\exp \pi(\zeta'-\zeta))\circ \sigma_{\varrho,1}.$$ such that $\tilde\Phi_{-1}$ take values in $P_\zeta^\tau$ and $\varphi$ is given up to isometry by $$\label{varphil}\varphi=\exp(\zeta'-\zeta)\tilde{\Phi}_{-1}\tau_0.$$ We conclude that, given a pair $(\zeta,\tau)$, where $\zeta\in \mathfrak{I}(SU(2n)^{\sigma_\varrho})$ is a $\varrho$-canonical element and $\tau$ is an outer involution of the form (\[invo1\]), any extended solution $\tilde{\Phi}:S^2\setminus D\to U_\zeta^\tau(SU(2n)))$ gives rise via (\[varphil\]) to an harmonic map $\varphi$ from the two-sphere into $\mathcal{L}^s_{2n}$ and, conversely, all harmonic two-spheres into $\mathcal{L}^s_{2n}$ arise in this way.

For $\mathcal{L}^s_{4}$, since $\exp\pi2\zeta_1$ belongs to the centre of $SU(4)$, we have five classes of harmonic maps into $\mathcal{L}^s_{4}$: $$(2\zeta_1,\sigma_{\varrho,1}),\, (\zeta_2,\sigma_{\varrho,1}),\,(2\zeta_1+\zeta_2,\sigma_{\varrho,1})\, (\zeta_2,\mathrm{Ad}\exp\pi\zeta_2\circ \sigma_{\varrho,1}),\,(2\zeta_1+\zeta_2,\mathrm{Ad}\exp\pi\zeta_2\circ\sigma_{\varrho,1}).$$

Let us consider in detail the class $(\zeta_2, \sigma_{\varrho,1})$. Clearly $r(\zeta_2)=2$. Let $W_1$, $W_2$, $W_3$ and $W_4$ be the complex one-dimensional images of $E_1$, $E_2$, $E_3$ and $E_4$, respectively. That is, $W_i=\mathrm{Span}\{v_i\}$, where $v_i$ are defined by (\[ves\]). Any extended solution $\Phi:S^2\setminus D\to U_{\zeta_2}^{\sigma_{\varrho,1}}$ is given by $\Phi=\exp C\cdot \gamma_{\zeta_2}$, with $\gamma_{\zeta_2}(\lambda)=\lambda^{-1}\pi_{W_4}+\pi_{W_3\oplus W_2 }+\lambda \pi_{W_1}$, for some holomorphic vector-valued function $C:S^2\setminus D\to (\mathfrak{u}^0_{\zeta_2})_{\sigma_{\varrho,1}}$, where $$(\mathfrak{u}^0_{\zeta_2})_{\sigma_{\varrho,1}}=(\mathfrak{p}_0^{\zeta_2})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}+\lambda(\mathfrak{p}_1^{\zeta_2})^\perp\cap
\mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,1}}$$ and $$\begin{aligned}
 (\mathfrak{p}_0^{\zeta_2})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}&=({\mathfrak{g}}_{L_1-L_2}\oplus{\mathfrak{g}}_{L_3-L_4} \oplus {\mathfrak{g}}_{L_1-L_3}  \oplus {\mathfrak{g}}_{L_2-L_4})\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}},   \\ (\mathfrak{p}_1^{\zeta_2})^\perp\cap\mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,1}}&= {\mathfrak{g}}_{L_1-L_4}\cap\mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,1}}= {\mathfrak{g}}_{L_1-L_4}.\end{aligned}$$

We have $\sigma_{\varrho,1}(X_{1,2})=-X_{3,4}$ and $\sigma_{\varrho,1}(X_{1,3})= X_{2,4}$. Hence we can write $C=C_0+C_1\lambda$, with $$C_0=a(X_{1,2}-X_{3,4})+b(X_{1,3}+X_{2,4}),\,\,\,\,\,  C_1=cX_{1,4}$$ for some meromorphic functions $a,b,c$ on $S^2$. The harmonicity equations impose that $ab'-ba'=0$, which means that $b=\alpha a$ for some constant $\alpha\in {\mathbb{C}}.$ Hence given arbitrary meromorphic functions $a,c$ on $S^2$ and a complex constant $\alpha$, $$\Phi=\left(
                                          \begin{array}{cccc}
                                            1 & a & \alpha a & c\lambda \\
                                            0 & 1 & 0 & -\alpha a \\
                                            0 & 0 & 1 & a \\ 0 & 0 & 0 & 1  \\
                                          \end{array}
                                        \right)\cdot \gamma_{\zeta_1},$$ is an extended solution associated to some harmonic map in the class $(\zeta_2, \sigma_{\varrho,1})$. Reciprocally, any harmonic map in such class arises in this way.

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