Solutions of Hitchin's self-duality equations correspond to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1995 asks whether all real sections give rise to global solutions of the self-duality equations. An armative answer would allow for complex analytic procedure to obtain solutions of the self- duality equations. The purpose of my talks is to explain the construction of 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., a rank 4 harmonic map theory, with the rank 2 self-duality theory.