  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.
