  A class of $ G $-invariant Einstein-Yang-Mills (EYM) systems with
cosmological constant on homogeneous spaces $ G / H $, where $ G $ is a
semisimple compact Lie group, is presented. These EYM--systems can be obtained
in terms of dimensional reduction of pure gravity. If $ G / H $ is a symmetric
space, the EYM--system on $ G / H $ provides a static solution of the
EYM--equations on spacetime $ {\Bbb R} \times G / H $. This way, in particular,
a solution for an arbitrary Lie group $ F $, considered as a symmetric space,
is obtained. This solution is discussed in detail for the case $ F = SU(2) $. A
known analytical EYM--system on $ {\Bbb R} \times S^3 $ is recovered and it is
shown - using a relation to the BPST instanton - that this solution is of
sphaleron type. Finally, a relation to the distance of Bures and to parallel
transport along mixed states is shown.
