Geodesic Flow on Three-Dimensional Ellipsoids with Equal Semi-Axes

Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semi-axes with $SO(2)$ symmetry, and ellipsoids with three semi-axes coinciding with $SO(3)$ symmetry. All of these cases are Liouville-integrable, and reduction of the symmetry leads to singular reduced systems on lower-dimensional ellipsoids. The critical values of the energy-momentum maps and their singular fibers are completely classified. In the cases with $SO(2)$ symmetry there are corank 1 degenerate critical points; all other critical points are non-degenreate. We show that in the case with $SO(2) \times SO(2)$ symmetry three global action variables exist and the image of the energy surface under the energy-momentum map is a convex polyhedron. The case with $SO(3)$ symmetry is non-commutatively integrable, and we show that the fibers over regular points of the energy-casimir map are $T^2$ bundles over $S^2$.