Integrable Systems on a Sphere, an Ellipsoid and a Hyperboloid

Affine transformations in Euclidean space generate a correspondence between integrable systems on cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant transformations, we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.