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We present a generalization of the spinor and twistor geometry for on(pseudo) Riemannian manifolds enabled with nonholonomic distributions or forFinsler-Cartan spaces modelled on tangent Lorentz bundles. Nonholonomic(Finsler) twistors are defined as solutions of generalized twistor equationsdetermined by spin connections and frames adapted to nonlinear connectionstructures. We show that the constructions for local twistors can be globalizedusing nonholonomic deformations with "auxiliary" metric compatible connectionscompletely determined by the metric structure and/or the Finsler fundamentalfunction. We explain how to perform such an approach in the Einstein gravitytheory formulated in Finsler like variables with conventional nonholonomic 2+2splitting.

Spinor and Twistor Geometry in Einstein Gravity and Finsler Modifications