I shall discuss the existence of infinite-dimensional invariant tori in a mechanical system made of infinitely many rotators weakly interacting with each other. I shall concentrate on interactions depending only on the angles, with the aim of discussing in a simple case the analyticity properties to be required on the perturbation of the integrable system in order to ensure the persistence of a large measure set of invariant tori with finite energy. Brindisi: al termine del seminario verrà servito un bicchiere di prosecco per festeggiare il terzo anniversario di LYSM.