An application of the continuity method for an equation on line bundles

A system of gauge equations, with 4 real parameters, is introduced on line bundles over closed Riemann surfaces. This system, similarly to the Vortex Equations, leads to a 2nd order non-linear elliptic equation that also appears on the problem of conformally pointwise curvature. A functional whose roots are the solutions to the elliptic equation is defined. Its partial derivative is invertible at a particular root. The Implicit Function Theorem yields a locally defined 1-parameter family of solutions. Uniqueness of the family is proven for a certain range of the parameter. The behaviour of this family beyond that range is discussed. A coordinate change on the original problem allows the construction of another functional, which is locally convex at its critical points, for a slightly larger range of the parameter.