Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

The paper deals with the study of the model of point vortices proposed by the German scientist Hermann Helmholtz. Necessary and sufficient conditions for the existence of infinitely many nonequivalent stationary configurations are found for a system consisting of two point vortices of intensity $\Gamma_1$ and an arbitrary number of point vortices of intensity $\Gamma_2$. A classification of such configurations is carried out. For the first time, a connection is discovered between the negative Diophantine Pell equation and stationary configurations of point vortices on the plane.