COMPATIBILITY OF SYMPLECTIC STRUCTURES ADAPTED TO NONCOMMUTATIVELY INTEGRABLE SYSTEMS

It is known that any integrable, possibly degenerate, Hamiltonian syst em is Hamiltonian relative to many different symplectic structures; un der certain hypotheses, the 'semi-local' structure of these symplectic forms, written in local coordinates of action-angle type, is also kno wn. The purpose of this paper is to characterize from the point of vie w of symplectic geometry the family of all these structures. The appro ach is based on the geometry of noncommutatively integrable systems an d extends a recent treatment of the nondegenerate case by Bogoyavlensk ij. Degenerate systems are comparatively richer in symplectic structur es than nondegenerate ones and this has the counterpart that the bi-Ha miltonian property alone does not imply integrability. However, integr ability is still guaranteed if a system is Hamiltonian with respect to three suitable symplectic structures. Moreover, some of the propertie s of recursion operators are retained. (C) 1998 Elsevier Science B.V.