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

TL;DR

This paper introduces a new family of compact integrable systems on  with two focus-focus singularities, expanding the class of explicit examples in semitoric systems and enabling future analysis of their invariants.

Contribution

It constructs a 6-parameter family of systems on  with two focus-focus points, filling a gap in explicit examples of semitoric systems.

Findings

System has exactly two focus-focus singularities.

Family is compact and integrable on .

Provides a basis for studying the twisting index invariant.

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.