A motion of fluid's free surface is considered in two dimensional (2D)
geometry. A time-dependent conformal transformation maps a fluid domain
into the lower complex half-plane of a new spatial variable. The fluid
dynamics is fully characterized by the motion of complex singularities
outside of fluid, i.e. in the upper complex half-plane, for the analytical
continuation of the conformal map and the complex velocity. Both a single
ideal fluid dynamics (corresponds e.g. to oceanic waves dynamics) and a
dynamics of superfluid Helium 4 with two fluid components are considered.
Both systems share the same type of the non-canonical Hamiltonian
structure. A superfluid Helium case is shown to be completely integrable
for the zero gravity and surface tension limit with the exact reduction to
the Laplace growth equation which is completely integrable through the
connection to the dispersionless limit of the integrable Toda hierarchy
and existence of the infinite set of complex pole solutions. A single
fluid case with nonzero gravity and surface tension turns more complicated
with the infinite set of new moving poles solutions found which are
however unavoidably coupled with the emerging moving branch points in the
upper half-plane. Residues of poles are the constants of motion. These
constants commute with each other in the sense of underlying non-canonical
Hamiltonian dynamics. It suggests that the existence of these extra
constants of motion provides an argument in support of the conjecture of
complete Hamiltonian integrability of 2D free surface hydrodynamics.