  We study the dynamics of quantized superfluid vortices on axisymmetric
compact surfaces with no holes, where the total vortex charge must vanish and
the condition of irrotational flow forbids distributed vorticity. A conformal
transformation from the surface to the complex plane allows us to use familiar
formalism to describe the motion of the quantized vortices and to find the
total energy. The simplest case is a vortex dipole with unit vortex charges on
an axisymmetric ellipsoid. We study two special symmetric vortex-dipole
configurations along with a general asymmetric one.
