Nijenhuis G-manifolds and Lenard bicomplexes: a new approach to KP systems

From the MR review by A.Sym: "One of the most important ingredients of the modern theory of integrable Hamiltonian systems in (1+1) dimensions is the concept of recursion operators. It is a general belief that this concept cannot be extended to the case of integrable Hamiltonian systems in (1+2) dimensions (such as the KP or Davey-Stewartson equations). The main goal of this paper is to correct this opinion. Indeed, the authors develop a new formal setting which is based on two important notions: (1) The Nijenhuis G-manifold and (2) their Lenard bicomplexes. Roughly speaking, the Nijenhuis G-manifold (G a Lie group) is a pair (M,G) of two Nijenhuis manifolds satisfying the following condition: G acts on M leaving its Nijenhuis structure (torsion-free tensor field of (1,1) type) invariant. This formal framework leads to a unified construction of recursion operators in (1+1) as well as in (1+2) dimensions. Some concrete realizations of this formal scheme (KdV and KP-hierarchies) are also discussed in the paper. "