  We model pseudo-Finsler geometries, with pseudo-Euclidean signatures of
metrics, for two classes of four dimensional nonholonomic manifolds: a) tangent
bundles with two dimensional base manifolds and b) pseudo-Riemannian/ Einstein
manifolds. Such spacetimes are enabled with nonholonomic distributions and
associated nonlinear connection structures and theirs metrics are solutions of
the field equations in general relativity or in generalized gravity theories
with nonholonomic variables. We rewrite the Schwarzschild metric in Finsler
variables and use it for generating new classes of locally anisotropic black
holes and (or) stationary deformations to ellipsoidal configurations. There are
analyzed the conditions when such metrics describe imbedding of black hole
solutions into nontrivial solitonic backgrounds.
