Geodesic flow on the ellipsoid with equal semi-axes

The equations for the 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 showed that the equations for the geodesic flow on the general ellipsoid with distinct semi-axes are Liouville-integrable, and described a set of integrals which weren't known classically. These integrals break down in the case of coinciding semi-axes. After reviewing the properties of the geodesic flow on the three-dimensional ellipsoid with distinct semi-axes, the three-dimensional ellipsoid with the two middle semi-axes being equal, corresponding to a Hamiltonian invariant under rotations, is investigated, using the tools of singular reduction and invariant theory. 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 map gives the bifurcation diagram. The fibres of the critical values of the energy momentum map are found, and an analysis is carried out of the action variables. The obstruction to the existence of single valued globally smooth action variables is monodromy. [Continues.]