What conserved quantity causes degeneracy of spectrum in rational polygonal billiards? 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.
 A: The case of square membrane is simple. Whenever the total Hamiltonian $\hat H$ is a sum of independent Hamiltonians for each direction, i.e.
$$\hat H=\hat H_x+\hat H_y,$$
both energies of the subsystems are conserved: $E_x$ and $E_y$. In the case of identical Hamiltonians for $x$ and $y$ we also have $x\leftrightarrow y$ exchange symmetry, i.e. if we consider the operator $\hat S$ such that
$$(\hat Sf)(x,y)=f(y,x),$$
then total Hamiltonian $\hat H$ commutes with it. But Hamiltonians $\hat H_x$ and $\hat H_y$ don't. For example, for $\hat H_x=-\frac1m\partial_x^2(\cdot)+U(x)(\cdot)$ we have
$$(\hat H_x\hat Sf)(x,y)=-\frac1m f^{(0,2)}(y,x)+U(x)f(y,x),$$
$$(\hat S\hat H_xf)(x,y)=-\frac1m f^{(2,0)}(y,x)+U(y)f(y,x),$$
$$\left(\left(\hat H_x\hat S-\hat S\hat H_x\right)f\right)(x,y)=\frac1m\left(f^{(2,0)}(y,x)-f^{(0,2)}(y,x)\right)+(U(x)-U(y))f(y,x).$$
So we have the following pairs of non-commuting conserved quantities:


*

*energy $E_x$ and $(x-y)$ parity $S$,

*energy $E_y$ and $(x-y)$ parity $S$.
Since all they individually commute with total Hamiltonian, there must be degeneracy in the spectrum. These conserved quantities exist not only for simple particle in a 2D box, but also for the case of $(x-y)Q$ potential, which separates into $Qx$ and $Qy$ parts (after coordinate change $y\to-y$), as well as for any other potential of the form $U(x)+U(y)$.
I guess that for general rectangular membrane one would have to somehow generalize $\hat S$ to explain the degeneracy in the case of rational sides ratio.
