Non-axisymmetric, scale-free, razor-thin discs

Galaxies exhibit a variety of non-axisymmetric structure (bars, spiralstructure, lopsided structure, etc.). These suggest the following generalproblem: what are the possible stationary configurations of a two-dimensionalself-gravitating fluid other than an axisymmetric razor-thin disc? We address amodest component of this problem: we seek non-axisymmetric razor-thin discs oftwo-dimensional barotropic fluid that are stationary in an inertial frame. Wedistinguish between `razor-thin' and `two-dimensional,' applying the latterterm to the equation of state. Furthermore we assume that our systems arescale-free, which reduces the partial differential equations describing thesystem to ordinary differential equations. We also allow for the presence of anaxisymmetric background potential. This simple and highly idealized problemalready exhibits a rich variety of solutions, the richest being for m=1symmetry.