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

We develop equations to describe the equilibrium state of scale-free, razor-thin barotropic fluid discs, with rotation curve v proportional to R(-beta), beta is an element of E(-1/4, 1/2). The discs may be embedded in a scale-free axisymmetric background potential. Nearly axisymmetric discs are constructed analytically, and discs with azimuthal density variations as large as 10:1 are constructed numerically. For small departures from axisymmetry, we find that (i) stationary self-consistent cold or cool discs with m greater than or equal to 2 do not exist; (ii) on a plane whose axes are the Mach number and the strength of the background potential, there are generally two one-parameter sequences of non-axisymmetric discs (one with aligned isodensity contours and one with spiral contours); the spiral sequence exists only for m=1; (iii) isolated discs with a flat rotation curve (beta=0) support non-axisymmetric equilibrium states at all Mach numbers, and discs with beta=1/4 support a two-parameter family of self-similar spiral patterns.