  We find new classes of exact solutions of the initial momentum constraint for
vacuum Einstein's equations. Considered data are either invariant under a
continuous symmetry or they are assumed to have the exterior curvature tensor
of a simple form. In general the mean curvature $H$ is non-constant and $g$ is
not conformally flat. In the generic case with the symmetry we obtain general
solution in an explicit form. In other cases solutions are given up to
quadrature. We also find a class of explicit solutions without symmetries which
generalizes data induced by the Kerr metric or other metrics related to the
Ernst equation. The conformal method of Lichnerowicz, Choquet-Bruhat and York
is used to prove solvability of the Hamiltonian constraint if $H$ vanishes.
Existence of marginally outer trapped surfaces in initial manifold is
discussed.
