On the affine invariant of simple hypersemitoric systems

Hypersemitoric systems are a class of integrable systems on $4$-dimensional symplectic manifolds which only have mildly degenerate singularities and where one of the integrals induces an effective Hamiltonian $S^1$-action and is proper. We introduce the affine invariant of hypersemitoric systems, which is a generalization of the Delzant polytope of toric systems and the polytope invariant of semitoric systems. Along the way, we compute and plot this invariant for meaningful and more and more complicated examples.