  Affine transformations in Euclidean space generates a correspondence between
integrable systems on cotangent bundles to the 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.
