  We show how the Einstein equations with cosmological constant (and/or various
types of matter field sources) can be integrated in a very general form
following the anholonomic deformation method for constructing exact solutions
in four and five dimensional gravity (S. Vacaru, IJGMMP 4 (2007) 1285). In this
letter, we prove that such a geometric method can be used for constructing
general non-Killing solutions. The key idea is to introduce an auxiliary linear
connection which is also metric compatible and completely defined by the metric
structure but contains some torsion terms induced nonholonomically by generic
off-diagonal coefficients of metric. There are some classes of nonholonomic
frames with respect to which the Einstein equations (for such an auxiliary
connection) split into an integrable system of partial differential equations.
We have to impose additional constraints on generating and integration
functions in order to transform the auxiliary connection into the Levi-Civita
one. This way, we extract general exact solutions (parametrized by generic
off-diagonal metrics and depending on all coordinates) in Einstein gravity and
five dimensional extensions.
