Integrable hierarchies and the modular class

Abstract

It is well-known that the Poisson-Nijenhuis manifolds, introduced by Kosmann-Schwarzbach and Magri form the appropriate setting for studying many classical integrable hierarchies. In order to define the hierarchy, one usually specifies in addition to the Poisson-Nijenhuis manifold a bi-hamiltonian vector field. In this paper we show that to every Poisson-Nijenhuis manifold one can associate a canonical vector field (no extra choices are involved!) which under an appropriate assumption defines an integrable hierarchy of flows. This vector field is the modular class of the Poisson-Nijhenhuis manifold. This class has a canonical representative which, under a cohomological assumption, is a bi-hamiltonian vector field. In many examples the associated hierarchy of flows reproduces classical integrable hierarchies. We illustrate in detail with the Harmonic Oscillator, the Calogero-Moser system, the classical Toda lattice and various Bogoyavlensky-Toda Lattices.