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We consider a particle moving on a cone and bound to its tip by $1/r$ orharmonic oscillator potentials. When the deficit angle of the cone divided by$2 \pi$ is a rational number, all bound classical orbits are closed.Correspondingly, the quantum system has accidental degeneracies in the discreteenergy spectrum. An accidental SU(2) symmetry is generated by the rotationsaround the tip of the cone as well as by a Runge-Lenz vector. Remarkably, someof the corresponding multiplets have fractional ``spin'' and unusualdegeneracies.

Runge-Lenz Vector, Accidental SU(2) Symmetry, and Unusual Multiplets for Motion on a Cone