Willmore surfaces in spheres via loop groups IV: on totally isotropic   Willmore two-spheres in $S^6$
  Totally isotropic surfaces in $S^6$ are not necessarily Willmore surfaces. Therefore it is the first goal of this paper to derive a geometric characterization of totally isotropic Willmore two-spheres in $S^6$. This will naturally yield to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, we also obtain an algorithm to construct totally isotropic Willmore two-spheres in $S^6$. As an application, a new, totally isotropic Willmore two-sphere which is not S-Willmore (i.e., has no dual surface) in $S^6$ is constructed, illustrating the theory of this paper. 
Peng Wang
math.DG