The factorisation method for constructing eigenstates and eigenvalues of the Schrödinger equation, based on the algebra of supersymmetry, is applied to second-order relativistic equations such as Kramers’ equation for the Dirac-Coulomb system (hydrogen atom); the well known helicity degeneracy, for example, of the \(2s_{}\) and \(2p_{}\) levels, which is broken by the Lamb shift, is thus associated with supersymmetry. A novel form of supersymmetry is found when \(l=0:\) two interpenetrating ladders, founded on different and non-degerate ’ground’ states, coexist. One ladder, corresponding to a deeply bound ground state, has no counterpart, in the physical hydrogen spectrum. Analogous results are obtained for the Klein-Gordon-Coulomb system in one and three dimensions. Eigenstates and eigenvalues for the one-dimensional Dirac-Coulomb system are found by projection.