  We consider Euclidean SU(N) Yang-Mills theory on the space GxR, where G is a
compact semisimple Lie group, and introduce first-order BPS-type equations
which imply the full Yang-Mills equations. For gauge fields invariant under the
adjoint G-action these BPS equations reduce to first-order matrix equations, to
which we give instanton solutions. In the case of G=SU(2)=S^3, our matrix
equations are recast as Nahm equations, and a further algebraic reduction to
the Toda chain equations is presented and solved for the SU(3) example.
Finally, we change the metric on GxR to Minkowski and construct finite-energy
dyon-type Yang-Mills solutions. The special case of G=SU(2)xSU(2) may be used
in heterotic flux compactifications.
