incompressible fluid wit h a free surface is considered in two dimensional< br> (2D) geometry. It is well known that the dynam ics of small to moderate

amplitudes of surface perturbations can be reformulated in terms of the

canonical Hamiltonian structure for the surfa ce elevation and Dirichlet

boundary condition of the velocity potential. Arbitrary large

per turbations can be efficiently characterized throug h a time-dependent

conformal mapping of a flui d domain into the lower complex half-plane. We

reformulate the exact Eulerian dynamics through a non-canonical nonlocal

Hamiltonian system for the pair of new conformal variables. The

corr esponding non-canonical Poisson bracket is non-deg enerate\, i.e. it

does not have any Casimir i nvariant. Any two functionals of the conformal

mapping commute with respect to the Poisson brack et. We also consider a

generalized hydrodynam ics for two components of superfluid Helium which< br> has the same non-canonical Hamiltonian structu re. In both cases the fluid

dynamics is fully characterized by the complex singularities in the upper

complex half-plane of the conformal map and the complex velocity.

Analytical continua tion through the branch cuts generically results i n the

Riemann surface with infinite number of sheets. An infinite family of

solutions with m oving poles are found on the Riemann surface. Resi dues of

poles are the constants of motion. The se constants commute with each other

in the se nse of underlying non-canonical Hamiltonian dynami cs which

provides an argument in support of th e conjecture of complete Hamiltonian

integrabi lity of surface dynamics. LOCATION:Seminar Room 2\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR