Lax operators, Poisson groups, and the differential Galois theory

We examine the transfer of a Poisson structure in a space of differential or difference Lax operators to the space of solutions of the corresponding auxiliary problem (the space of wave functions) in detail. We investigate a spontaneous symmetry breaking resulting in the appearance of a nontrivial Poisson structure on the differential or difference Galois group. We review the difference version of the Drinfeld–Sokolov theory and describe a new type of classical $r$-matrices related to generalized exchange algebras.