Action-angle Coordinates for Integrable Systems on Poisson Manifolds

We prove the action-angle theorem in the general context of non-commutative integrable systems on Poisson manifolds. It implies the existence of action-angle coordinates for Liouville integrable systems on Poisson manifolds, a generalization of the classical result, which is stated in the context of symplectic manifolds. The proof uses the calculus of polyvector fields, rather than differential forms, in particular, we use in the end a Poisson version of the classical Carathéodory–Jacobi–Lie theorem, which we also prove.