Action-angle variables are also important in obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. They only exist, providing a key characterization of the dynamics, when the system is completely integrable, i.e.,  the number of independent Poisson commuting invariants is maximal and the conserved energy surface is compact.
Action-angle variables are significant in determining the frequencies of oscillatory or rotational motion without requiring the solution of the equations of motion. This occurs when the system is completely integrable, where the number of independent Poisson commuting invariants is maximized and the conserved energy surface is compact.