We construct a one-parameter family \(F_t=(J, H_t)_{0 \le t \le 1}\) of integrable systems on a compact 4-dimensional symplectic manifold \((M, \omega )\) that changes smoothly from a toric system \(F_0\) with eight elliptic–elliptic singular points via toric type systems to a semitoric system \(F_t\) for \( t^-< t < t^+\). These semitoric systems \(F_t\) have precisely four elliptic–elliptic and four focus–focus singular points. Moreover, at \(t= \frac{1}{2}\), the system has precisely two focus–focus fibres each of which contains exactly two focus–focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.

我们在紧凑的 4 维辛流形\((M, \omega )\)上构造了一个可积系统的单参数族\(F_t=(J, H_t)_{0 \le t \le 1}\)从具有八个椭圆-椭圆奇异点的复曲面系统\（F_0 \）通过复曲面类型系统平稳地转换为\（t ^-<t <t ^ + \）的半复曲面系统\（F_t \）。这些符号系统\（F_t \）恰好具有四个椭圆-椭圆和四个焦点-焦点奇异点。此外，在\(t= \frac{1}{2}\)，该系统具有精确的两个聚焦-聚焦纤维，每个纤维都包含两个精确的聚焦-焦点，使这些纤维具有双捏环的形状。我们示例性地明确参数化这些纤维之一。