We study theoretically a potential motion of fluid with a free boundary and with no external forces. We derive and integrate the equations describing the flow evolution from the initial to a highly nonlinear stage in 2D and 3D cases and study the influence of the initial conditions in a wide range of parameters. It is shown that at late time a nonlinear structure of bubbles and spikes is generated and the structure is determined by the initial velocity field. The local dynamics of highly symmetric 3D flows has a universal form when expressed in dimensionless units. Bubbles and spikes conserve a near-circular contour, and at a fixed length scale, the 3D solutions depend significantly on the flow symmetry. We show that the influence of the initial conditions could result in nonuniqueness of solutions describing the regular bubble. (C) 2000 American Institute of Physics. [S1070-6631(00)00812-6].