Integrable generalisations of Dirac magnetic monopole

TL;DR

This paper classifies integrable quantum and classical generalizations of the Dirac magnetic monopole on a sphere, identifying two families with quadratic integrals, including a new family involving elliptic functions.

Contribution

It completes the classification of integrable monopole generalizations on $S^2$, introducing a new family characterized by elliptic functions and special metrics.

Findings

Two integrable families with quadratic integrals identified

Classical family corresponds to Dirac monopole in harmonic electric field

New family involves elliptic functions on $S^2$ with special metrics

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.