In this paper we study conservative systems describing the motion of a particle on the line in the field of potential force with additional quasiperiodic time-dependence.
We show that the superquadratic growth of the potential at infinity results in the near-integrability of the Hamiltonian system in question (for a large class of potentials), despite the fact that no smallness assumptions are made on the quasiperiodic dependence of the potential on time. As a consequence all the solutions of such systems are bounded for all time.
Some specific examples are given, together with a counterexample shich shows that, without the quasiperiodicity assumption, the boundedness breaks down.