Vu Ngocs Conjecture on focus-focus singular fibers with multiple pinched points

We classify, up to symplectomorphisms, a neighborhood of a singular fiber of an integrable system (which is proper and has connected fibers) containing $k > 1$ focus-focus critical points. Our result shows that there is a one-to-one correspondence between such neighborhoods and $k$ formal power series, up to a $(mathbb{Z}_2 times D_k)$-action, where $D_k$ is the $k$-th dihedral group. The $k$ formal power series determine the dynamical behavior of the Hamiltonian vector fields $X_{f_1}, X_{f_2}$ associated to the components $f_1, f_2 colon (M, omega) to mathbb{R}$ of the integrable system on the symplectic manifold $(M,omega)$ via the differential equation $omega(X_{f_i}, cdot) = mathop{}!mathrm{d} f_i$, near the singular fiber containing the $k$ focus-focus critical points. This proves a conjecture of San Vu Ngoc from 2002.