Symmetry of magnetized self-gravitating hydrodynamic equation

We deal with a polytropic gas model for the gravitational collapse of a magnetized self-gravitating isothermal cylinder. We concentrate on radial distribution and restrict a magnetic field to a purely toroidal field. In this study, we carry out a symmetry analysis to construct an exact solution of the model and present a group invariant solution with the magnetic field given as an arbitrary function. We consider two specific cases for the description of the magnetic field. In the first case, the magnetic flux passes through a volume, which is proportional to the mass in it. In the second case, the ratio of the gas pressure to magnetic pressure is constant. In both cases, the solutions are similar to Emden solution.