$1/r$ or $r^2$ potential [@Has07]. When the deficit angle of the cone is a rational fraction of $2 \pi$, again all bound classical orbits are closed and there are accidental degeneracies in the energy spectrum of the quantum system. In this case the Runge-Lenz vector does not act as a self-adjoint operator in the domain of the Hamiltonian. Remarkably, as a consequence of this unusual property, the accidental $SU(2)$ symmetry has unusual multiplets with fractional (i.e. neither integer nor half-integer) spin.

An interesting example of an accidental symmetry involving a vector potential is cyclotron motion [@Lan30; @Joh49]. Also in this case, there is a deep connection between the fact that all bound classical orbits are closed and additional degeneracies in the discrete energy spectrum of the corresponding quantum system. As was already noted in [@Joh49], the center of the circular cyclotron orbit is a conserved quantity analogous to the Runge-Lenz vector in the Kepler problem. Also the radius of the cyclotron orbit is a conserved quantity directly related to the energy. Interestingly, while the two coordinates of the center are not simultaneously measurable, the radius of the circle has a sharp value in an energy eigenstate. In the cyclotron problem, 