  We apply the method of moving anholonomic frames, with associated nonlinear
connections, in (pseudo) Riemannian spaces and examine the conditions when
various types of locally anisotropic (la) structures (Lagrange, Finsler like
and more general ones) could be modeled in general relativity. New classes of
solutions of the Einstein equations with generic local anisotropy are
constructed. We formulate the theory of nearly autoparallel (na) maps and
introduce the tensorial na-integration as the inverse operation to both
covariant derivation and deformation of connections by na-maps. The problem of
redefinition of the Einstein gravity theory on na-backgrounds, provided with a
set of na-map invariant conditions and local conservation laws, is analyzed.
There are illustrated some examples of generation of vacuum Einstein fields by
Finsler like metrics and chains of na-maps.
