Exact solutions of the Euler equations for some two-dimensional incompressible flows

Two-dimensional (plane and axisymmetric) steady flows of an ideal incompressible fluid are considered in a potential field of external forces. An elliptic partial differential equation is obtained such that each of its solutions is a stream function of a flow described by a certain solution of the Euler equations. Examples of such new exact solutions are given. These solutions can be used, in particular, for testing numerical algorithms and computer programs.