  We consider the motion of ideal incompressible fluid with free surface. We
analyzed the exact fluid dynamics though the time-dependent conformal mapping
$z=x+iy=z(w,t)$ of the lower complex half plane of the conformal variable $w$
into the area occupied by fluid. We established the exact results on the
existence vs. nonexistence of the pole and power law branch point solutions for
$1/z_w$ and the complex velocity. We also proved the nonexistence of the
time-dependent rational solution of that problem for the second and the first
order moving pole.
