  We study Willmore surfaces of constant Moebius curvature $K$ in $S^4$. It is
proved that such a surface in $S^3$ must be part of a minimal surface in $R^3$
or the Clifford torus. Another result in this paper is that an isotropic
surface (hence also Willmore) in $S^4$ of constant $K$ could only be part of a
complex curve in $C^2\cong R^4$ or the Veronese 2-sphere in $S^4$. It is
conjectured that they are the only examples possible. The main ingredients of
the proofs are over-determined systems and isoparametric functions.
