Higher solutions of Hitchin's self-duality equations

Solutions of Hitchin's self-duality equations corresponds to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for complex analytic procedure to obtain all solutions of the self-duality equations. The purpose of this paper is to construct counter examples given by certain (branched) Willmore surfaces in $3$-space (with monodromy) via the generalized Whitham flow. Though these higher solutions do not give rise to global solutions of the self-duality equations on the whole Riemann surface $M$, they are solutions on an open dense subset of it. This suggest a deeper connection between Willmore surfaces, i.e., rank $4$ harmonic maps theory, with the rank $2$ self-duality theory.