Willmore surfaces are the extremals of the Willmore functional (possibly
under a constraint on the conformal structure). With the characterization of
Willmore surfaces by the (possibly perturbed) harmonicity of the mean curvature
sphere congruence [Blaschke, Ejiri, Rigoli, Burstall-Calderbank], a
zero-curvature formulation follows [Burstall-Calderbank]. Deformations on the
level of harmonic maps prove to give rise to deformations on the level of
surfaces, with the definition of a spectral deformation
[Burstall-Pedit-Pinkall, Burstall-Calderbank] and of a Baecklund transformation
[Burstall-Quintino] of Willmore surfaces into new ones, with a Bianchi
permutability between the two [Burstall-Quintino]. This text is dedicated to a
self-contained account of the topic, from a conformally-invariant viewpoint, in
Darboux's light-cone model of the conformal n-sphere.Comment: v2: some extra detail added, 35 page