A lowest order constrained variational method for calculating the binding energy of nuclear matter, previously proposed by the present authors, is extended to treat the strong tensor force components of realistic NN potentials. Numerical results are given for three early potentials of Gammel, Christian and Thaler, and reasonable agreement is found with a previous calculation of Ristig, Ter Louw and Clark which included three-body cluster contributions to the energy. A range of five potentials giving good fits to the experimental two-body NN data is also studied, and binding energies of typically 22 MeV per nucleon at saturation densities corresponding to k F ≈ 1.6–1.7 fm −1, are found. For three of the potentials considered, comparison is made with the recent results of Pandharipande and Wiringa, which include the contributions to the energy from all of the most significant many-body clusters, and excellent agreement is found. It is suggested that explicit inclusion of some of the neglected internal degrees of freedom of the nucleons, such as the possibility of excitation to Δ (1236) states, might bring the equilibrium nuclear matter results closer to the empirical values.