Previously, we showed that surface density profiles of the form of a power-law times a Sérsic function satisfy the hydrostatic Jeans equations, a variety of observational constraints, and the condition of a minimal radial entropy profile in two-dimensional galaxy discs with fixed power law, halo potentials. It was assumed that such density profiles are generated by star scattering by clumps, waves, or other inhomogeneities. Here, we generalize these models to self-gravitating discs. The cylindrically symmetric Poisson equation imposes strong constraints. Scattering processes favour smoothness, so the smoothest solutions, which minimize entropy gradients, are preferred. In the case of self-gravitating discs (e.g. inner discs), the gravity, surface density, and radial velocity dispersion in these smoothest models are all of the form 1/r times an exponential. When vertical balance is included, the vertical velocity dispersion squared has the same form as the surface density, and the scale height is constant. In combined self-gravitating plus halo gravity cases, the radial dispersion has an additional power-law term. None the less, the surface density profile has the same form at all radii, without breaks, satisfying the ‘disc–halo conspiracy’. The azimuthal velocity and velocity dispersions are smooth, though the former can have a distinct peak. In these models the vertical dispersion increases inwards, and scattering may mediate a transition to a secular bulge. If halo gravity dominates vertically in the outer disc, it flares. The models suggest a correlation between disc mass and radial scale length. The combination of smoothness, simplicity, ability to match generic observational features, and physical constraints is unique to these models.