The breakdown of a line vortex

The conditions giving rise to a singularity in solutions to quasi-cylindrical approximation equations for a line vortex are investigated. The breakdown of the vortex is treated as a transition from the supercritical state to the subcritical state at lengths of the order of the vortex radius. The conditions under which the singularity and the criticality criteria are the same are determined. A classification of possible singularities is proposed, and it is shown that in the general case, with a given longitudinal external pressure gradient, the solution in the vicinity of the singularity cannot be continued beyond the singular point.