Compatibility of symplectic structures adapted to noncommutatively integrable systems

It is known that any integrable, possibly degenerate, Hamiltonian system is Hamiltonian relative to many different symplectic structures; under certain hypotheses, the ‘semi-local’ structure of these symplectic forms, written in local coordinates of action-angle type, is also known. The purpose of this paper is to characterize from the point of view of symplectic geometry the family of all these structures. The approach is based on the geometry of noncommutatively integrable systems and extends a recent treatment of the nondegenerate case by Bogoyavlenskij. Degenerate systems are comparatively richer in symplectic structures than nondegenerate ones and this has the counterpart that the bi-Hamiltonian property alone does not imply integrability. However, integrability is still guaranteed if a system is Hamiltonian with respect to three suitable symplectic structures. Moreover, some of the properties of recursion operators are retained.