  Previously, we showed that surface density profiles of the form of a
power-law times a Sersic 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 favor 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. Nonetheless, 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.
