The geometry of soliton equations

From the MR review by R.Schmid: "Ideas from the geometrical study of soliton equations are used to give an explanation of some algorithmic procedures used in the field of the inverse scattering technique (IST). In the first three sections particular classes of manifolds are introduced: Poisson manifolds, Poisson-Nijenhuis manifolds and GN manifolds, which give the natural framework for the study of equations solvable by the IST. A hierarchy of GN manifolds is constructed in Section 5. In the next section the reduction technique is developed on these manifolds. It is shown that many algorithms of IST are realizations of the projection technique naturally associated to the geometry of GN manifolds. A detailed study of the Veselov-Novikov system is given in Section 7. The last sections deal with the construction of the symmetry algebras and associated Lenard bicomplexes and their reductions. It is shown that different approaches in the literature are simply different realizations of this scheme. "