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