Geometric structure of "broadly integrable" Hamiltonian systems

We study the geometry of the fibration in invariant tori of a Hamiltonian system which is integrable in Bogoyavlenskij’s “broad sense”—a generalization of the standard cases of Liouville and non-commutative integrability. We show that the structure of such a fibration generalizes that of the standard cases. Firstly, the base manifold has a Poisson structure. Secondly, there is a natural way of arranging the invariant tori which generates a second foliation of the phase space; however, such a foliation is not just the polar to the invariant tori. Finally, under suitable conditions, there is a notion of an “action manifold” with an affine structure. We also study the analogous of the problem of the existence of “global action-angle coordinates” for these systems.