  We prove an almost global in time existence result of small amplitude space
periodic solutions of the 1D gravity-capillary water waves equations with
constant vorticity. The result holds for any value of gravity, vorticity and
depth and any surface tension belonging to a full measure set. The proof
demands a Hamiltonian paradifferential Birkhoff normal form reduction for
quasi-linear PDEs in presence of resonant wave interactions: the normal form
may be not integrable but it preserves the Sobolev norms thanks to its
Hamiltonian nature. A major difficulty is that usual paradifferential calculus
used to prove local well posedness (as the celebrated Alinhac good unknown)
does not preserve the Hamiltonian structure. A major novelty of this paper is
to develop an algorithmic perturbative procedure \`a la Darboux to correct
usual paradifferential transformations to symplectic maps, up to an arbitrary
degree of homogeneity. The symplectic correctors turn out to be smoothing
perturbations of the identity, and therefore only slightly modify the
paradifferential structure of the equations. The Darboux procedure which
recovers the nonlinear Hamiltonian structure is written in an abstract
functional setting, in order to be applicable also in other contexts.
