Classical degeneracy and the existence of additional constants of motion

The relation between degeneracy of classical frequencies in integrable systems and additional global constants of motion is investigated from a general point of view. It is shown that in autonomous systems with N degrees of freedom the existence of Λ<N such additional invariants imply Λ global linear relations with commensurate coefficients between the frequencies, and vice versa. Furthermore, such a degeneracy gives rise to a specific structure of the Hamiltonian considered as a function of action variables. The general statements are illustrated by various examples