Superintegrable Hamiltonian systems: Geometry and perturbations

Many and important integrable Hamiltonian systems are 'superintegrable', in the sense that there is an open subset of their 2d-dimensional phase space in which all motions are linear on tori of dimension n < d. A thorough comprehension of these systems requires a description which goes beyond the standard notion of Liouville - Arnold integrability, that is, the existence of an invariant fibration by Lagrangian tori. Instead, the natural object to look at is formed by both the fibration by the ( isotropic) invariant tori and by its (coisotropic) polar foliation, which together form what in symplectic geometry is called a 'dual pair', or 'bifoliation', or 'bifibration'. We review this geometric structure, relating it to the dynamical properties of superintegrable systems and pointing out its importance for a thorough understanding of these systems.