A compact semi-toric system on a compact four dimensional symplectic manifold can be thought of as a Hamiltonian $S^1 \times \mathbb{R}$-action whose singular orbits are non-degenerate in a symplectic Morse-Bott sense. Introduced by Vu Ngoc and classified under a mild genericity assumption by Pelayo and Vu Ngoc, these systems lie at the crossroads of completely integrable Hamiltonian systems, symplectic toric manifolds and Hamiltonian $S^1$-spaces (by restricting the action to $S^1$). This talk studies the relation between compact semi-toric systems and Hamiltonian $S^1$-spaces, the latter classified by Karshon, with two motivating questions:

- What is the minimal set of invariants of compact semi-toric systems needed to reconstruct those of the underlying Hamiltonian $S^1$-space?
- Which Hamiltonian $S^1$-spaces arise as those underlying a compact semi-toric system?

The aim of the talk is to provide an answer to Question 1 above (which is joint work with S. Hohloch and S. Sabatini); if time permits, there will be some comments to illustrate how to go about tackling Question 2 (which is joint work with S. Hohloch, S. Sabatini and M. Symington).

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