The zero curvature form of integrable hierarchies in the Z x Z-matrices

In this paper it is shown how one can associate to a finite number of commuting directions in the Lie algebra of upper triangular Z X Z-matrices an integrable hierarchy consisting of a set of evolution equations for perturbations of the basic directions inside the mentioned Lie algebra. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent with zero curvature equations for a collection of finite band matrices, that are the components of a formal connection form. One concludes with the linearization of the hierarchies and the notion of wave matrices at zero, which is the algebraic substitute for a basis of the horizontal sections of the formal connection corresponding to this connection form.