Exactness of Bohr-Sommerfeld quantisation for two non-central potentials

In this paper we demonstrate the integrability of the Hamilton-Jacobi equation for two non-central potentials in spherical polar coordinates, and present complete solutions for the classically bound orbits. We then show that the semiclassical method of Bohr-Sommerfeld quantisation exactly reproduces the bound state spectra of the corresponding quantum mechanical Schr\"odinger equations. One of these potentials has previously been analysed in parabolic coordinates; the results for the other are, to the authors' best knowledge, original.