  We prove that the Einstein equations can be solved in a very general form for
arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases
following a geometric method of anholonomic frame deformations for constructing
exact solutions in gravity. The main idea of this method is to introduce on
(pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection)
metric compatible linear connection which is also completely defined by the
same metric structure. Such a canonically distinguished connection is with
nontrivial torsion which is induced by some nonholonomy frame coefficients and
generic off-diagonal terms of metrics. It is possible to define certain classes
of adapted frames of reference when the Einstein equations for such an
alternative connection transform into a system of partial differential
equations which can be integrated in very general forms. Imposing nonholonomic
constraints on generalized metrics and connections and adapted frames
(selecting Levi-Civita configurations), we generate exact solutions in Einstein
gravity and extra dimension generalizations.
