A complexity-one space on a symplectic 2n-manifold is the Hamiltonian action of a torus of dimension n-1. The momentum map for such an action can be identified with a collection of n-1 real valued functions. On the other hand, an integrable system on such a manifold is the data of n functions. This motivates several natural questions: given a complexity-one space, when can an additional function be found to produce an integrable system? When can the resulting system be chosen to be toric? When can it be chosen to have other nice properties, such as having no degenerate singularities?

The case of when a circle action on a 4-manifold can be lifted to a toric integrable system was already completely understood by Karshon in 1999, but most other cases have remained open until relatively recently, such as the semitoric case, studied by Hohloch-Sabatini-Sepe-Symington. In this talk, I will discuss answers to various versions of these questions, both in dimension four and higher. Parts of this work are joint with Sonja Hohloch, Susan Tolman, and Jason Liu.