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

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