[en] A new, non adiabatic, description of the H₂⁺ molecular ion and its isotopomers is proposed: the molecular system is treated as a three body Coulomb system, in the framework of non relativistic quantum mechanics. The method takes advantage of the dynamical symmetries of the system. It relies on the choice of the perimetric coordinates to describe the system, and of a generalized Hylleraas basis, in which the Hamiltonian exhibits strong coupling rules. The method is described in detail both for S and P states (i.e., states of total angular momentum J=0 or 1). We calculate the energies of all J=0 or 1 vibrational levels of the H₂⁺ and D₂⁺ molecular ions with a very high accuracy (typically 10^-14 atomic unit). This a considerable improvement over previous calculations. The dependence of these results on the proton to electron mass ratio is also discussed. (orig.)