Semitoric systems of non-simple type

A semitoric integrable system $F=(J,H)$ on a symplectic $4$-manifold is simple if each fiber of $J$ contains at most one focus-focus point. Simple semitoric systems were classified about ten years ago in terms of five invariants. In the first part of the paper we explain how the simplicity assumption can be removed from the classification by adapting the invariants. In the second part of the paper we reformulate the classification in terms of the piecewise integral affine group of the plane, which offers an alternative way to understand the invariants of semitoric systems.