Black Holes, Ellipsoids, and Nonlinear Waves in Pseudo-Finsler Spaces and Einstein Gravity

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.