  Hypersemitoric systems are 2-degree-of-freedom integrable systems on
4-dimensional manifolds that have an underlying $S^1$-symmetry and no
degenerate singularities apart from maybe a finite number of families of
so-called parabolic singularities. We give a short overview of recent examples
displaying various bifurcations and sketch a topological-combinatorial
classification of the connected components of fibers of hypersemitoric systems.
