Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vũ Ngọc about a decade ago in terms of five symplectic invariants. If a semitoric system has several focus–focus singularities, then some of these invariants have multiple components, one for each focus–focus singularity. Their computation is not at all evident, especially in multi-parameter families. In this paper, we consider a four-parameter family of semitoric systems with two focus–focus singularities. In particular, apart from the polygon invariant, we compute the so-called height invariant. Moreover, we show that the two components of this invariant encode the symmetries of the system in an intricate way.

Semitoric系统是一类特殊的完全可积分系统，具有两个自由度，大约在十年前，Pelayo和VũNgọc根据五个辛不变式对它们进行了分类。如果Semitoric系统具有多个焦点-焦点奇点，那么其中一些不变量具有多个分量，每个焦点-焦点奇点一个。它们的计算一点也不明显，尤其是在多参数系列中。在本文中，我们考虑具有两个焦点-焦点奇点的半参数系统的四参数系列。特别是，除了多边形不变式之外，我们还计算所谓的高度不变式。此外，我们证明了该不变式的两个组成部分以复杂的方式编码系统的对称性。