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 \(k \geqslant 1\) 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 \((\mathbb {Z}_2 \times D_k)\)-action, where \(D_k\) is the kth 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 \((M,\omega )\) near the singular fiber containing the k focus-focus critical points. This proves a conjecture of San Vũ Ngọc from 2003.

我们对包含\(k \geqslant 1\)个焦点-焦点临界点的可积系统（其图像正确且具有连接的纤维）的奇异纤维的饱和邻域进行分类，直到纤维辛同胚。我们的证明恢复了本文之前已知的\(k=1\)分类。我们的结果表明，这样的邻域和k 个形式幂级数之间存在一一对应，最多为\((\mathbb {Z}_2 \times D_k)\) -action，其中\(D_k\)为第k个二面体群。k个形式幂级数决定了与包含k 个焦点-焦点临界点的奇异纤维附近的辛流形\((M,\omega )\)上的动量图分量相关的哈密顿矢量场的动力学行为。这证明了San Vũ Ngọc 2003年的猜想。