After reviewing the properties of the 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 symmetry reduction leads
to a (singular) system on a two-dimensional ellipsoid with an additional potential
and with a hard billiard wall inserted in the middle coordinate plane. We show that
the regular part of the image of the energy momentum map is not simply connected
and there is an isolated critical value for zero angular momentum. The singular
fiber of the isolated singular value is a doubly pinched torus multiplied by a circle.
This circle is not a group-orbit of the symmetry group, and thus analysis of this
fiber is non-trivial. Finally we show that the system has a non-trivial monodromy,
and consequently does not admit single valued globally smooth action variables.