Integrable systems with symmetries: Toric, semitoric, and beyond

This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable systems called semitoric families, and explain how deforming systems through controlled bifurcations in such families (and their generalizations) can be used to construct explicit semitoric systems with prescribed invariants. The first part of the paper serves as a quick introduction to integrable systems for newcomers to the field, such as graduate students, while the majority of the exposition surveys recent developments and technical details that will be of interest to experts. It closes with a look at future directions, including hypersemitoric systems and complexity one integrable systems.