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

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