Restricted Integrability and Arnold Liouville Integrability

Consider a Hamiltonian H in n degrees of freedom, admitting functionally independent Poisson commuting first integrals J_1,..,J_k. When k=n, this system is called Arnold Liouville integrable, and on compact common levels, the dynamic is quasi periodic. When k<n, let us further assume that the dynamical system restricted to the smooth common zero level S of these first integrals admit additional first integrals K_1,..,K_{n-k}. We will show that the property ``Poisson commuting'' can be properly defined for these additional first integrals. What can be said about the dynamic on S?

We will prove that the system is always integrable by quadrature. For k=1, the compact common levels are always tori, and under a Diophantine condition, the motion is quasi periodic. For k=2, n=3, we will prove that a compact common level N is an orientable torus bundle over a circle. It is either a 3-torus, a nilmanifold, or there exists an additional first integral whose level are 2-tori. In the 3 torus case, under a Diophantine condition, the motion is quasi periodic, in the 2 torus case, the dynamic is quasi periodic but with non global action angles coordinates over N. In the nilmanifold case, the system has a parabolic dynamic on it. We will then present the construction explicit of examples and obstructions.