  We use a systematic construction method for invariant connections on
homogeneous spaces to find the Einstein-SU(n)-Yang-Mills equations for
Friedmann-Robertson-Walker and locally rotationally symmetric homogeneous
cosmologies. These connections depend on the choice of a homomorphism from the
isotropy group into the gauge group. We consider here the cases of the gauge
group SU(n) and SO(n) where these homomorphisms correspond to unitary or
orthogonal representations of the isotropy group. For some of the simpler cases
the full system of the evolution equations are derived, for others we only
determine the number of dynamical variables that remain after some mild fixing
of the gauge.
