Abstract: Motivated by Gromov's non-squeezing theorem and Viterbo's volume-capacity conjecture, a central (hard!) question in symplectic topology is to study the orbit of a ball under the action of the group of symplectomorphisms of the standard symplectic vector space. A special instance of this question is to understand which Lagrangian products are symplectic balls, where a Lagrangian product is the product of a set in configuration space with a set in the space of momenta. In this talk we provide some answers to the latter question: for instance, the Lagrangian product of an equilateral triangle and a regular hexagon is symplectomorphic to a ball in dimension four. To prove these results we use the Toda lattice, its relation to billiards and ideas from integrable Hamiltonian systems. This is joint work with V. G. Ramos and Y. Ostrover.