Alonso, JaumeHohloch, Sonja2021-07-122021-07-122021-04-190938-8974https://depositonce.tu-berlin.de/handle/11303/13413http://dx.doi.org/10.14279/depositonce-12197Semitoric 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.en510 Mathematikcompletely integrable Hamiltonian systemssemitoric systemssymplectic invariantsheight invariantfocus-focus singularitiesArticle1432-1467