  In this paper we construct new solutions of the Kahler-Yang-Mills equations,
by applying dimensional reduction methods to the product of the complex
projective line with a compact Riemann surface. The resulting equations, that
we call gravitating vortex equations, describe Abelian vortices on the Riemann
surface with back reaction of the metric. As a particular case of these
gravitating vortices on the Riemann sphere we find solutions of the
Einstein-Bogomol'nyi equations, which physically correspond to Nielsen-Olesen
cosmic strings in the Bogomol'nyi phase. We use this to provide a Geometric
Invariant Theory interpretation of an existence result by Y. Yang for the
Einstein-Bogomol'nyi equations, applying a criterion due to G. Szekelyhidi.
