  A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented
conformal Gauss map $Gr_y:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$,
which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$.
An easy observation shows that all conformal Gauss maps of Willmore surfaces
satisfy a restricted nilpotency condition which will be called "strongly
conformally harmonic." The goal of this paper is to characterize those strongly
conformally harmonic maps from a Riemann surface $M$ to $SO^+ (1, n +
3)/{SO^+(1, 3) \times SO(n) }$ which are the conformal Gauss maps of some
Willmore surface in $S^{n+2}.$ It turns out that generically the condition of
being strongly conformally harmonic suffices to be associated to a Willmore
surface. The exceptional case will also be discussed.
