Off-Diagonal Deformations of Kerr Black Holes in Einstein and Modified Massive Gravity and Higher Dimensions

We find general parameterizations for generic off-diagonal spacetime metrics and matter sources in general relativity, GR, and modified gravity theories when the field equations decouple with respect to certain types of nonholonomic frames of reference. This allows us to construct various classes of exact solutions when the coefficients of fundamental geometric/ physical objects depend on all spacetime coordinates via corresponding classes of generating and integration functions and/or constants. Such (modified) spacetimes can be with Killing and non-Killing symmetries, describe nonlinear vacuum configurations and effective polarizations of cosmological and interaction constants. Our method can be extended to higher dimensions which simplifies some proofs for imbedded and nonholonomically constrained four dimensional configurations. We reproduce the Kerr solution and show how to deform it nonholonomically into new classes of generic off-diagonal solutions depending on 3-8 spacetime coordinates. There are analyzed certain examples of exact solutions determined by contributions of new type of interactions and sources in massive gravity and/or modified f(R,T) gravity. We conclude that considering generic off-diagonal nonlinear parametric interactions in GR it is possible to mimic various effects in massive and/or modified gravity, or to distinguish certain classes of "generic" modified gravity solutions which can not be encoded in GR