Toric domains are a special class of symplectic manifolds in C^n, invariant by the standard circle action in each of the one-dimensional complex planes. This abundance of symmetries makes the study of embedding problems into toric domains and computations of symplectic capacities of toric domains, often possible. So when considering these kinds of problems in general manifolds (for example: disk cotangent bundles of surfaces), a strategy could be to produce a toric domain out of your given manifold and "reduce" the original problem to a similar one in toric domains.
In this talk, I will explain how to carry out this idea for computing the biggest ball that can be symplectically embedded into the disk cotangent bundle of an ellipsoid of revolution. The idea to obtain such a related toric domain comes by studying an integrable system for the disk cotangent bundle of an ellipsoid of revolution and using Arnol'd-Liouville Theorem to obtain action-angle coordinates. We then use the obtained toric domain to suggest a candidate to the best symplectically embedded ball. Finally, to show that this is, as a matter of fact, the best possible ball, we use some obstructional tools, more specifically, ECH capacities. This is joint work with Brayan Ferreira and Vinicius Ramos.