This question arises from the comments relevant to the post When is the ergodic hypothesis reasonable?
Consider a Hamiltonian system having more effective degrees of freedom than conserved quantities. For example, let's assume that the number of degrees of freedom is $D=3$ and that the only conserved quantity is the Hamiltonian $H$ itself. In whatever constant-energy surface of the type $H=E_0$, there will be both chaotic and regular regions. Consider a regular trajectory in phase space. The relevant motion consists in regular oscillations, so it should be possible to conveniently introduce action / angle variables.
1) Am I wrong?
Variables of the type "action" are conserved quantities.
2) Isn't it strange? Where do they come from? It seems to me that a non-integrable system can feature regular regimes where additional conserved quantities are present. Even stranger, these conserved quantities break as soon as one considers trajectories outside the regular islands.