  We show that a geometric techniques can be elaborated and applied for
constructing generic off-diagonal exact solutions in $f(R,T)$--modified gravity
for systems of gravitational-Yang-Mills-Higgs equations. The corresponding
classes of metrics and generalized connections are determined by generating and
integration functions which depend, in general, on all space and time
coordinates and may possess, or not, Killing symmetries. For nonholonomic
constraints resulting in Levi-Civita configurations, we can extract solutions
of the Einstein-Yang-Mills-Higgs equations. We show that the constructions
simplify substantially for metrics with at least one Killing vector. There are
provided and analyzed some examples of exact solutions describing generic
off-diagonal modifications to black hole/ellipsoid and solitonic
configurations.
