  Recently, classical results on completeness of trajectories of Hamiltonian
systems obtained at the beginning of the seventies, have been revisited,
improved and applied to Lorentzian Geometry. Our aim here is threefold: to give
explicit proofs of some technicalities in the background of the specialists, to
show that the introduced tools allow to obtain more results for the
completeness of the trajectories, and to apply these results to the
completeness of spacetimes that generalize classical plane and pp-waves.
