Department of Mathematics, University of Alberta
Edmonton, Canada T6G~2G1
The singular boundary value problem that arises for the static
spherically symmetric SU(n)-Einstein-Yang-Mills
equations in the so-called magnetic case is analyzed.
Among the possible actions of SU(2) on a SU(n)-principal bundles
over space-time there is one which appears to be the most natural.
If one assumes that no electrostatic type component is present in
the Yang-Mills fields and the gauge is suitably fixed a set of n-1
second order and two first order differential equations is obtained
for n-1 gauge potentials and two metric components as functions of
the radial distance. This system generalizes the one for the case
n=2 that lead to the discrete series of the Bartnik-Mckinnon and
the corresponding black hole solutions. It is highly nonlinear and
singular at r=infinity and at