A classical problem of the dynamics of the free surface of an ideal incompressible fluid with infinite depth has been considered. It has been found that the regime of motion of the fluid where the pressure is a quadratic function of the velocity components is possible in the absence of external forces and capillarity. It has been shown that equations of plane potential flow for this situation are linearized in conformal variables and are then easily solved analytically. The found solution includes an arbitrary function specifying the initial shape of the surface of the fluid. The developed approach makes it possible for the first time to locally describe the formation of various singularities on the surface of the fluid—bubbles, drops, and cusps.