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Superintegrable Lissajous systems on the sphere

A kind of systems on the sphere, whose trajectories are similar to the Lissajous curves, are studied by means of one example. The symmetries are constructed following a unified and straightforward procedure for both the quantum and the classical versions of the model. In the quantum case it is stressed how the symmetries give the degeneracy of each energy level. In the classical case it is shown how the constants of motion supply the orbits, the motion and the frequencies in a natural way.