  The equations for geodesic flow on the ellipsoid are well known, and were
first solved by Jacobi in 1838 by separating the variables of the
Hamilton-Jacobi equation. In 1979 Moser investigated the case of the general
ellipsoid with distinct semi-axes and described a set of integrals which
weren't know classically. After reviewing the properties of geodesic flow on
the three dimensional ellipsoid with distinct semi-axes, we investigate the
three dimensional ellipsoid with the two middle semi-axes being equal,
corresponding to a Hamiltonian invariant under rotations. The system is
Liouville-integrable and thus the invariant manifolds corresponding to regular
points of the energy momentum map are 3-dimensional tori. An analysis of the
critical points of the energy momentum maps gives the bifurcation diagram. We
find the fibres of the critical values of the energy momentum map, and carry
out an analysis of the action variables. We show that the obstruction to the
existence of single valued globally smooth action variables is monodromy.
