  The classification of Willmore 2-spheres in the $n$-dimensional sphere $S^n$
is a long-standing problem, solved only when $n=3,4$ by Bryant, Ejiri, Musso
and Montiel independently. In this paper we give a classification when $n=5$.
There are three types of such surfaces up to M\"obius transformations: (1)
super-conformal surfaces in $S^4$; (2) minimal surfaces in $R^5$; (3) adjoint
transforms of super-conformal minimal surfaces in $R^5$. In particular,
Willmore surfaces in the third class are not S-Willmore (i.e., without a dual
Willmore surface).
  To show the existence of Willmore 2-spheres in $S^5$ of type (3), we describe
all adjoint transforms of a super-conformal minimal surfaces in $R^n$ and
provide some explicit criterions on the immersion property. As an application,
we obtain new immersed Willmore 2-spheres in $S^5$ and $S^6$, which are not
S-Willmore.
