Two topics in integrable Hamiltonian systems

Francesco Fasso, Universita` di Padova and UC Santa Cruz

The talk is devoted to some foundational questions about the characterization of integrable Hamiltonian systems. One is the inadequacy of the notion of complete integrability to account for the global structure of the fibration by invariant tori of degenerate systems (e.g., the free rigid body, Kepler); a thorough comprehension of these systems should instead be based on the so called `non--commutative integrability', which focuses on the finest fibration by invariant tori. The second question concerns the existence of systems which are integrable (in the sense that all their motions are quasi--periodic) but not completely (nor noncommutatively) integrable.