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There are described equations for a pair comprising a Riemannian metric and aKilling field on a surface that contain as special cases the Einstein Weylequations (in the sense of D. Calderbank) and a real version of a special caseof the Abelian vortex equations, and it is shown that the property that ametric solve these equations is preserved by the Ricci flow. The equations aresolved explicitly, and among the metrics obtained are all steady gradient Riccisolitons (e.g. the the cigar soliton) and the sausage metric; there are foundother examples of eternal, ancient, and immortal Ricci flows, as well as someRicci flows with conical singularities.

Ricci flows on surfaces related to the Einstein Weyl and Abelian vortex equations