Magnetic impurities, integrable vortices and the Toda equation

The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong–Wong type. Under certain conditions, these generalizations remain integrable. We further set up a gauge theory with product group, two complex scalar fields and general charge matrix. second species vortices, when frozen, interpreted as impurity for all equations. then give geometric compatibility condition, which enables us remove constant term in This is similar reduction from Taubes equation Liouville equation. find family matrices that turn equations into either Toda or opposite sign. exact analytic solutions cases solution sign appears be new.