  This paper is concerned with giving the proof that there is a general
decoupling property of vacuum and nonvacuum gravitational field equations in
Einstein gravity and $f(R,T)$-modifications. The constructions are possible in
terms of geometric and physical objects adapted to certain nonholonomic 2+2
splitting with local fibred structure. This allows us to generate exact and/or
parametric depending solutions with generic off-diagonal metrics and
generalized, or Levi-Civita, connections. Different classes of modified
spacetimes are determined by corresponding generating and integration functions
depending, in general, on all space and time coordinates and may possess, or
not, Killing symmetries. The initial data sets for the Cauchy problem and their
global properties are analyzed. There are formulated the criteria of evolution
with spacetime splitting and decoupling of fundamental field equations.
Examples of exact solutions defining ellipsoid deformations of black hole
metrics and solitonic configurations are provided.
