  On a 4-dimensional compact symplectic manifold, we study how suitable
perturbations of a toric system to a family of completely integrable systems
with $\mathbb{S}^1$-symmetry lead to various hyperbolic-regular singularities.
We compute and visualise associated phenomena like flaps, swallowtails, and
$k$-stacked tori for $k \in \{2, 3, 4\}$ and give an upper bound for $k$ in our
family of systems.
