Integrable almost--symplectic Hamiltonian systems

We extend the notion of Liouville integrability, which is peculiar to Hamiltonian systems on symplectic manifolds, to Hamiltonian systems on almost-symplectic manifolds, namely, manifolds equipped with a nondegenerate (but not closed) 2-form. The key ingredient is to require that the Hamiltonian vector fields of the integrals of motion in involution (or equivalently, the generators of the invariant tori) are symmetries of the almost-symplectic form. We show that, under this hypothesis, essentially all of the structure of the symplectic case (from quasiperiodicity of motions to an analog of the action-angle coordinates and of the isotropic-coisotropic dual pair structure characteristic of the fibration by the invariant tori) carries over to the almost-symplectic cas