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We classify, up to fiberwise symplectomorphisms, a saturated neighborhood ofa singular fiber of an integrable system (which is proper onto its image andhas connected fibers) containing $k > 1$ focus-focus critical points. Ourresult shows that there is a one-to-one correspondence between suchneighborhoods and $k$ formal power series, up to a $(\mathbb{Z}_2 \timesD_k)$-action, where $D_k$ is the $k$-th dihedral group. The $k$ formal powerseries determine the dynamical behavior of the Hamiltonian vector fieldsassociated to the components of the momentum map on the symplectic manifold$(M,\omega)$ near the singular fiber containing the $k$ focus-focus criticalpoints. This proves a conjecture of San Vu Ngoc from 2002.

Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points