 Anisotropic quantum harmonic oscillator

For a positive diagonal quadratic potential $V=m{\sum }_i{\omega }_i^2{x}_i^2/2$, coordinate Hamiltonians commute and wavefunctions separate into one-dimensional oscillator factors. Equality of two squared frequencies gives rotation symmetry in their plane; equality of all three gives full rotation symmetry. Equalities create energy degeneracies and extra conserved angular momentum. Merely finding an accidental equality between particular level energies does not establish a global rotational symmetry.