Symplectic classification of hypersemitoric systems

Develop a complete symplectic classification of four-dimensional hypersemitoric integrable systems (M, ω, F = (J, H)), in which J generates an effective S^1-action and all degenerate singularities are of parabolic type. The objective is to determine a full set of invariants that classify such systems up to isomorphism, extending the classification frameworks known for toric and semitoric systems.

Background

Hypersemitoric systems generalize semitoric systems by allowing mildly degenerate (parabolic) singularities and requiring one integral to generate a proper effective S^1-action. While toric systems are classified by Delzant polytopes and semitoric systems by a set of five invariants including a polytope invariant, a comprehensive classification for hypersemitoric systems has not yet been achieved.

This paper introduces an affine invariant for hypersemitoric systems—generalizing the Delzant and semitoric polytope invariants—and computes representatives for various examples. However, the work does not provide a full classification, highlighting the need for a systematic framework that completely characterizes hypersemitoric systems up to symplectic isomorphism.