  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.
