The Percus-Yevick (PY) and the hypernetted chain (HNC) integral equations have been applied to realistic, non-spherical, intermolecular potentials for nitrogen. Non-additive contributions, in the form of the Axilrod-Teller three-body interaction, have been considered. The integral equations were solved after expansion in spherical harmonics and the solution procedure is described in detail. Thermodynamic properties of the gas calculated through the compressibility and the virial routes are in surprisingly good agreement with experimental data. Typical deviations are within +/-0.2 per cent for both p rho T and sound-speed data at sub-critical densities. Some improvement may be achieved by fitting the three-body dispersion coefficient. We conclude that a simple integral-equation theory and a reasonably sophisticated anisotropic pair potential, coupled with the Axilrod-Teller three-body term, can offer a good description of the thermodynamic surface at sub-critical densities. A closure more accurate than either PY or HNC will be required if good results are to be obtained at densities above the critical.