  An analytic solution for an uniaxial spherical resonator is presented using
the method of Debye potentials. This serves as a starting point for the
calculation of whispering gallery modes (WGM) in such a resonator. Suitable
approximations for the radial functions are discussed in order to best
characterize WGMs. The characteristic equation and its asymptotic expansion for
the anisotropic case is also discussed, and an analytic formula with a
precision of the order $O[\nu^{-1}]$ is also given. Our careful treatment of
both boundary conditions and asymptotic expansions makes the present work a
particularly suitable platform for a quantum theory of whispering gallery
resonators.
