A family of compact semitoric systems with two focus-focus singularities

Abstract

About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on ${\mathbb{S}}^2\times {\mathbb{R}}^2$ and coupled angular momenta on ${\mathbb{S}}^2\times {\mathbb{S}}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on ${\mathbb{S}}^2\times {\mathbb{S}}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.

Update to most recent version: some typos removed; minor inaccuracies corrected; better layout