  The semiclassical treatment of the two-dimensional harmonic oscillator
provides an instructive example of the relation between classical motion and
the quantum mechanical energy spectrum. We extend previous work on the
anisotropic oscillator with incommensurate frequencies and the isotropic
oscillator to the case with commensurate frequencies for which the Lissajous
curves appear as classical periodic orbits. Because of the three different
scenarios depending on the ratio of its frequencies, the two-dimensional
harmonic oscillator offers a unique way to explicitly analyze the role of
symmetries in classical and quantum mechanics.
