Superintegrable Hamiltonian system

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a formula_1-dimensional symplectic manifold for which the following conditions hold:
(i) There exist formula_2 independent integrals formula_3 of motion. Their level surfaces (invariant submanifolds) form a fibered manifold formula_4 over a connected open subset formula_5.
(ii) There exist smooth real functions formula_6 on formula_7 such that the Poisson bracket of integrals of motion reads
formula_8. 
(iii) The matrix function formula_6 is of constant corank formula_10 on formula_7.
If formula_12, this is the case of a completely integrable Hamiltonian system. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville-Arnold theorem on action-angle coordinates of completely integrable Hamiltonian system as follows.
Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold formula_13 is a fiber bundle
in tori formula_14. There exists an open neighbourhood formula_15 of formula_13 which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates formula_17,
formula_18, formula_19 such that formula_20 are coordinates on formula_14. These coordinates are the Darboux coordinates on a symplectic manifold formula_15. A Hamiltonian of a superintegrable system depends only on the action variables formula_23 which are the Casimir functions of the coinduced Poisson structure on formula_24.
The Liouville-Arnold theorem for completely integrable systems and the Mishchenko-Fomenko theorem for the superintegrable ones are generalized to the case of non-compact invariant submanifolds. They are diffeomorphic to a toroidal cylinder formula_25.