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Totally isotropic surfaces in $S^6$ are not necessarily Willmore surfaces.Therefore it is the first goal of this paper to derive a geometriccharacterization of totally isotropic Willmore two-spheres in $S^6$. This willnaturally yield to a description of such surfaces in terms of the loop grouplanguage. Moreover, applying the loop group method, we also obtain an algorithmto construct totally isotropic Willmore two-spheres in $S^6$. As anapplication, a new, totally isotropic Willmore two-sphere which is notS-Willmore (i.e., has no dual surface) in $S^6$ is constructed, illustratingthe theory of this paper.

Willmore surfaces in spheres via loop groups IV: on totally isotropic Willmore two-spheres in $S^6$