S. Ruijsenaars and his school introduced integrable dynamical systems of relativistic invariance (classical and quantic). A Lax pair approach was latter proposed by M. Bruschi and F. Calogero (CMP, 85). In this talk, I will report on a recent joint work with F. Calogero in which we introduce a new presentation of these systems based on the associated second-order differential equation. We solve a conjecture posed last year by F. Calogero (to appear in JMP) on the solution of certain perturbations of these systems with completely periodic trajectories.