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

We summarize some our recent results on encoding exact solutions of field equations in Einstein and modified gravity theories into solitonic hierarchies derived for nonholonomic curve flows with associated bi-Hamilton structure. We argue that there is a canonical distinguished connection for which the fundamental geometric/ physical equations decouple in general form. This allows us to construct very general classes of generic off-diagonal solutions determined by corresponding types of generating and integration functions depending on all (spacetime) coordinates. If the integral varieties are constrained to zero torsion configurations, we can extract solutions for the general relativity theory. We conclude that the geometric and physical data for various classes of effective/modified Einstein spaces can be encoded into multi-component versions of the sine-Gordon, or modified Korteweg - de Vries equations.