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We summarize some our recent results on encoding exact solutions of fieldequations in Einstein and modified gravity theories into solitonic hierarchiesderived for nonholonomic curve flows with associated bi-Hamilton structure. Weargue that there is a canonical distinguished connection for which thefundamental geometric/ physical equations decouple in general form. This allowsus to construct very general classes of generic off-diagonal solutionsdetermined by corresponding types of generating and integration functionsdepending on all (spacetime) coordinates. If the integral varieties areconstrained to zero torsion configurations, we can extract solutions for thegeneral relativity theory. We conclude that the geometric and physical data forvarious classes of effective/modified Einstein spaces can be encoded intomulti-component versions of the sine-Gordon, or modified Korteweg - de Vriesequations.

Generic Off--Diagonal Solutions and Solitonic Hierarchies in Einstein and Modified Gravity