Past exam of the classical mechanics course of Part B of the physics course of the University of Oxford / 2016 / 4 / d / Solution

An autonomous Hamiltonian system with $n$ degrees of freedom is Liouville integrable if it has $n$ conserved functions $I_1,\dots ,I_n$, including the Hamiltonian, whose differentials are independent on a regular open set and which are in involution:The equations can then be reduced locally to quadratures. The statement about action-angle coordinates on invariant tori additionally requires compact connected regular common level sets; integrable unbounded motion need not be periodic or lie on a torus.