  We discuss the implementation, to the case of compact manifolds, of the
perturbative method of Friedrich-Butscher for the construction of solutions to
the vaccum Einstein constraint equations. This method is of a perturbative
nature and exploits the properties of the extended constraint equations ---a
larger system of equations whose solutions imply a solution to the Einstein
constraints. The method is applied to the construction of nonlinear
perturbations of constant mean curvature initial data of constant negative
sectional curvature. We prove the existence of a neighbourhood of solutions to
the constraint equations around such initial data, with particular components
of the extrinsic curvature and electric/magnetic parts of the spacetime Weyl
curvature prescribed as free data. The space of such free data is parametrised
explicitly.
