 Here snh is the hyperbolic analogue of sn, given by
and formula_27 and formula_28 are Jacobi elliptic functions of modulus formula_23.The associated transfer matrix formula_30 thus is a function of formula_25 alone; for all formula_25, formula_33

The other crucial part of the solution is the existence of a nonsingular matrix-valued function formula_36, such that for all complex formula_25 the matrices formula_38 commute with each other and the transfer matrices, and satisfy

where

The existence and commutation relations of such a function are demonstrated by considering pair propagations through a vertex, and periodicity relations of the theta functions, in a similar way to the six-vertex model.The commutation of matrices in () allow them to be diagonalised, and thus eigenvalues can be found.