  The Lax representations of the geodesic flow, the Jacobi-Rosochatius problem
and its perturbations by means of separable polynomial potentials, on a
ellipsoid are constructed. We prove complete integrability in the case of a
generic symmetric ellipsoid and describe analogous systems on complex
projective spaces. Also, we consider billiards within an ellipsoid under the
influence of the Hook and Rosochatius potentials between the impacts. A
geometric interpretation of the integrability analogous to the classical
Chasles and Poncelet theorems is given.
