2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/137771We 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.16 pages. Mistakes occured in the proof to the main theorem (Thm 3.6) has been correctedDifferential Geometry53A30 (53C21, 53C24, 53C42)text