Central potentials with closed cruise orbits

A particle will be said to be in cruise motion if it is nonholonomically constrained to have constant speed. When the particle is placed in a central force field, the resulting mechanical system is known to be integrable. Cruise orbits in a central force field may be closed (periodic in time) or not, depending on the potential and on the speed. Here we give a constructive characterization of all central potentials for which all cruise motions of a given speed are closed. As a consequence, we also give a new proof of the fact that for any such potential the set of speeds for which all cruise motions are closed has always empty interior.