Integrable generalisations of Dirac magnetic monopole

Abstract: We classify certain integrable (both classical and quantum) generalisations of Dirac magnetic monopole on topological sphere $S^2$ with constant magnetic field, completing the previous local results by Ferapontov, Sayles and Veselov. We show that there are two integrable families of such generalisations with integrals, which are quadratic in momenta. The first family corresponds to the classical Clebsch systems, which can be interpreted as Dirac magnetic monopole in harmonic electric field. The second family is new and can be written in terms of elliptic functions on sphere $S^2$ with very special metrics.