Local obstructions to a conformally invariant equation on M\"obius surfaces

On a M"obius surface, as defined by D. Calderbank, we study a variant of the Einstein-Weyl (EW) equation which we call scalar-flat M"obius EW (sf-MEW). This is a conformally invariant, finite type, overdetermined system of semi-linear partial differential equations. We derive local algebraic constraints for this equation to admit a solution and give local obstructions. In the generic case when a certain invariant of the M"obius structure given by a symmetric tensor $M_{ab}$ is non-zero, the obstructions are given by resultants of 3 polynomial equations whose coefficients are conformal invariants of the M"obius structure. The vanishing of the resultants is a necessary condition for there to be solutions to sf-MEW.