  With a number of special Hamiltonians, solutions of the Schr\"{o}dinger
equation may be found by separation of variables in more than one coordinate
system. The class of potentials involved includes a number of important
examples, including the isotropic harmonic oscillator and the Coulomb
potential. Multiply separable Hamiltonians exhibit a number of interesting
features, including "accidental" degeneracies in their bound state spectra and
often classical bound state orbits that always close. We examine another
potential, for which the Schr\"{o}dinger equation is separable in both
cylindrical and parabolic coordinates: a $z$-independent $V\propto
1/\rho^{2}=1/(x^{2}+y^{2})$ in three dimensions. All the persistent, bound
classical orbits in this potential close, because all other orbits with
negative energies fall to the center at $\rho=0$. When separated in parabolic
coordinates, the Schr\"{o}dinger equation splits into three individual
equations, two of which are equivalent to the radial equation in a Coulomb
potential---one equation with an attractive potential, the other with an
equally strong repulsive potential.
