  We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an
infinite family of solvable and integrable quantum systems in the plane,
indexed by the positive parameter k. Key components of their analysis were to
demonstrate that there are closed orbits in the corresponding classical system
if k is rational, and for a number of examples there are generating quantum
symmetries that are higher order differential operators than two. Indeed they
conjectured that for a general class of potentials of this type, quantum
constants of higher order should exist. We give credence to this conjecture by
showing that for an even more general class of potentials in classical
mechanics, there are higher order constants of the motion as polynomials in the
momenta. Thus these systems are all superintegrable.
