Apart from Laplace-Runge-Lenz vector conservation in Coulombic and something similar in harmonic and other central potentials, something leads to existence of periodic trajectories in such systems as rational polygonal billiards (although not all trajectories are periodic)$^\dagger$.
I believe in quantum case and in the case of vibrating membrane this results in degeneracy such that e.g. for a square membrane non-standing-wave eigenmodes are possible, like these:
This degeneracy isn't lifted even in the presence of a particular form of linearly-changing potential, see e.g. this question.
So I suppose there must be some quantity, which is conserved in such billiards. Its quantum operator, as I understand, should be the generator of the wave motions presented above (like momentum operator is the generator of translations, angular momentum operator is the generator of rotations, etc.).
So what is this conserved quantity which leads to periodic trajectories?
$^\dagger$ In fact, after some thinking I'm not sure that periodic trajectories have anything to do with this. They are possible even in irrational-side-ratio rectangular billiard, it's just that their projections of velocities also have irrational ratio to make rationally-related periods of oscillation in different directions.