Steady vortex flows of a self-gravitating gas

An exact solution of equations of steady motion of a self-gravitating gas is found. This solution describes a vortex flow of the gas from the surface of a spherical source and is partly invariant with respect to the group of rotations (Ovsyannikov vortex, singular vortex). The factor-system of the solution is reduced to finite formulas and one ordinary differential equation of the third order. Various regimes of gas motion described by this solution are determined: unlimited spreading of the gas with swirling from the surface of a spherical source and gas exhaustion with formation of a sphere with elevated density at a finite distance from this source.