Non-Abelian Vortices on Riemann Surfaces: an Integrable Case
  We consider U(n+1) Yang-Mills instantons on the space \Sigma\times S^2, where \Sigma is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n+1) instanton equations on \Sigma\times S^2 are equivalent to non-Abelian vortex equations on \Sigma. Solutions to these equations are given by pairs (A,\phi), where A is a gauge potential of the group U(n) and \phi is a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g>1, when \Sigma\times S^2 becomes a gravitational instanton, the non-Abelian vortex equations are the compatibility conditions of two linear equations (Lax pair) and therefore the standard methods of integrable systems can be applied for constructing their solutions. 
Alexander D. Popov