We show that all compact four-dimensional Hamiltonian $S^1$-spaces can be extended to a completely integrable system on the same manifold such that all singularities are non-degenerate, except possibly for a finite number of degenerate orbits of parabolic (also called cuspidal) type -- we call such systems hypersemitoric. More precisely, given any compact four dimensional Hamiltonian $S^1$-space $(M,omega,J)$ we show that there exists a smooth $Hcolon Mtomathbb{R}$ such that $(M,omega,(J,H))$ is a completely integrable system of hypersemitoric type. Hypersemitoric systems generalize semitoric systems. In addition to elliptic-elliptic, elliptic-regular, and focus-focus singular points which can occur in semitoric systems, hypersemitoric systems may also have hyperbolic-regular and hyperbolic-elliptic singular points (hyperbolic-hyperbolic points cannot appear due to the presence of the global $S^1$-action) and moreover degenerate singular points of a relatively tame type called parabolic. Admitting the existence of degenerate points is necessary since there exist compact four-dimensional Hamiltonian $S^1$-spaces whose extensions must include degenerate singular points of some kind as we show in the present paper. Parabolic points are among the most common and natural degenerate points, and we show that it is sufficient to only admit these degenerate points in order to extend all Hamiltonian $S^1$-spaces. In this sense, hypersemitoric systems are thus the nicest and smallest class of systems to which all Hamiltonian $S^1$-spaces can be extended. Moreover, we prove several foundational results about these systems, such as the non-existence of loops of hyperbolic-regular points and properties about their fibers.
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