  We present a new initial data formulation to solve the full set of Einstein
equations for spacetimes that contain a black hole under general conditions.
The method can be used to construct complete initial data for spacetimes (the
full metric) that contain a black hole. Contrary to most current studies the
formulation requires minimal assumptions. For example, rather than imposing the
form of the spatial conformal metric we impose 3 gauge conditions adapted to
the coordinates describing the system under consideration. For stationary,
axisymmetric spacetimes our method yields Kerr-Schild black holes in vacuum and
rotating equilibrium neutron stars. We demonstrate the power of our new method
by solving for the first time the whole system of Einstein equations for a
nonaxisymmetric, self-gravitating torus in the presence of a black hole. The
black hole has dimensionless spin $J_{\rm bh}/M_{\rm bh}^2=0.9918$, a rotation
axis tilted at a $30^\circ$ angle with respect to the angular momentum of the
disk, and a mass of $\sim 1/5$ of the disk.
