F. Fasso et T. Ratiu, COMPATIBILITY OF SYMPLECTIC STRUCTURES ADAPTED TO NONCOMMUTATIVELY INTEGRABLE SYSTEMS, Journal of geometry and physics, 27(3-4), 1998, pp. 199-220
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.