  We study the existence of infinite-dimensional invariant tori in a mechanical
system of infinitely many rotators weakly interacting with each other. We
consider explicitly 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. The proof we provide of
the persistence of the invariant tori implements the Renormalization Group
scheme based on the tree formalism -- i.e. the graphical representation of the
solutions of the equations of motion in terms of trees -- which has been widely
used in finite-dimensional problems. The method is very effectual and flexible:
it naturally extends, once the functional setting has been fixed, to the
infinite-dimensional case with only minor technical-natured adaptations.
