  We review the fundamentals of coupling constant metamorphosis (CCM) and the
St\"ackel transform, and apply them to map integrable and superintegrable
systems of all orders into other such systems on different manifolds. In
general, CCM does not preserve the order of constants of the motion or even
take polynomials in the momenta to polynomials in the momenta. We study
specializations of these actions which do preserve polynomials and also the
structure of the symmetry algebras in both the classical and quantum cases. We
give several examples of non-constant curvature 3rd and 4th order
superintegrable systems in 2 space dimensions obtained via CCM, with some
details on the structure of the symmetry algebras preserved by the transform
action.
