Invariant incompatible Poisson structures

Abstract

Supplemental invariant Poisson structures P c which are incompatible with the original Poisson structure P 1 are discovered for an arbitrary completely integrable Hamiltonian system. For the non-degenerate case, the complete classification of the invariant Poisson structures P c is obtained provided that the invariant submanifolds of the integrable Hamiltonian system are compact. The instability of the property of compatibility of any supplemental invariant non-degenerate Poisson structure P 2 with P 1 is established.