Deforming commuting directions in the space of Z × Z-matrices

In this work we start from various basic sets of commuting directions in the Z × Z-matrices that are the generators of a group of commuting flows. The main topic in this thesis form deformations of these original directions w.r.t. the parameters of these flows and their evolution equations. The combination of the evolutions of the perturbed matrices w.r.t. all directions yields so-called hierarchies of nonlinear differential and difference equations. Specific examples of these equations that one gets in this way, are the equations satisfied by infinite Toda lattices. We study three types of deformations: the first deforms the basic directions into the lower triangular matrices with the leading term equal to the basic direction. The second type of deformation is into the upper triangular matrices, where one drops the preservation of the leading term. In the third case one deforms roughly speaking half of the directions according to the first category and the other half according to the second one. In all three instances we study the algebraic structure of the equations and show the equivalence of the various forms in which they occur. One of them, the zero curvature form, is an indication that there is a relation with an infinite dimensional Cauchy problem. We show the uniqueness of the solvability of this Cauchy problem in the formal power series setting. Finally, the discussion of each type of deformation is concluded with a geometric construction of solutions of the hierarchies. This furnishes new illustrative examples of infinite dimensional varieties that play a central role at integrable hierarchies