Summary: It is shown that when a dynamical system \({\mathbf X}_0\) with a proper set of global first integrals is perturbed, the phase space region accessible to the orbits of the perturbed vector field \({\mathbf X}_0+{\mathbf X}_p\) is bounded (the authors are assuming here that the time variable runs over a finite interval). A polynomial new bound is obtained for the separation between the solutions of \({\mathbf X}_0\) and \({\mathbf X}_0+{\mathbf X}_p\). Perturbations near an equilibrium point of \({\mathbf X}_0\) are also considered.