[en] The solutions of a covariant two-body equation are applied to the spectra of H, muonium and positronium. In order to compare with the standard results we have expanded the potentials, the recoil terms and the perturbation calculations up to order α⁵. The best known intervals, namely Hfs of H, muonium and positronium and the 2³S₁ - 1³S₁ interval of positronium are obtained to this order in agreement with standard results and experiments. Having thus verified the nonperturbative equation to order α⁵, the recoil and normal and anomalous magnetic moment terms can now be evaluated numerically to all orders. (author)