General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations

Over the last seventy years, many Finsler-type geometric and modified gravity theories have been elaborated. They have been formulated in terms of different classes of Finsler generating functions, metric and nonmetric structures, nonlinear and linear connections, and various sets of postulated fundamental geometric objects with corresponding nonholonomic dynamical or evolution equations. In several approaches, the resulting gravitational and matter field equations were not completely defined geometrically, or were developed only for restricted models. We present a progress report with historical remarks and a summary of new results on Finsler - Lagrange - Hamilton geometric flow and gravity theories. Such theories can be constructed in an axiomatic form on (co) tangent Lorentz bundles as nontrivial modifications of Einstein gravity.