Vũ Ngọc’s conjecture on focus-focus singular fibers with multiple pinched points

We classify, up to fiberwise symplectomorphisms, a saturated neighborhood of a singular fiber of an integrable system (which is proper onto its image and has connected fibers) containing k ⩾ 1 focus-focus critical points. Our proof recovers the classification for k = 1 which was known prior to this paper. Our result shows that there is a one-to-one correspondence between such neighborhoods and k formal power series, up to a ( Z 2 × D k ) -action, where D k is the k th dihedral group. The k formal power series determine the dynamical behavior of the Hamiltonian vector fields associated to the components of the momentum map on the symplectic manifold ( M , ω ) near the singular fiber containing the k focus-focus critical points. This proves a conjecture of San Vũ Ngọc from 2003.