The last three equations can be recognized to be the equations of the form d μ μ = ν ν , d ν ν = 0 where d is the exterior derivative on the space of differential forms in the Z-space. Such equations can be solved with the help of the Poincare Lemma. In addition one needs to know how to multiply by the Klein operator from the right, which is easy to derive from the integral formula for the star-product: