  We consider a particle moving on a cone and bound to its tip by $1/r$ or
harmonic 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 discrete
energy spectrum. An accidental SU(2) symmetry is generated by the rotations
around the tip of the cone as well as by a Runge-Lenz vector. Remarkably, some
of the corresponding multiplets have fractional ``spin'' and unusual
degeneracies.
