A Family of Semitoric Systems with Four Focus-Focus Singularities and Two Double Pinched Tori

We construct a one-parameter family F-t = (J, H-t)(0 <= t <= 1) of integrable systems on a compact 4-dimensional symplectic manifold (M, omega) that changes smoothly from a toric system F-0 with eight elliptic-elliptic singular points via toric type systems to a semitoric system F-t for t(-) < t < t(+). These semitoric systems F-t have precisely four elliptic-elliptic and four focus-focus singular points. Moreover, at t = 1/2, the system has precisely two focus-focus fibres each of which contains exactly two focus-focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.

Automated summary

A one-parameter family of integrable systems on a compact 4-dimensional symplectic manifold is studied, transitioning smoothly from a system with elliptic-elliptic singular points to a semitoric system with a specific distribution of singular points. At t = 1/2, the system exhibits two focus-focus fibres with a unique shape.