Transformations of harmonic bundles and Willmore surfaces

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 harmonicity of the mean curvature sphere congruence ([Ejiri], [Rigoli]), a zero-curvature formulation follows ([Burstall and Calderbank]). Deformations on the level of bundles prove to give rise to deformations on the level of surfaces, with the definition of a spectral deformation ([Burstall and Calderbank]) and of a B\"{a}cklund transformation ([Burstall and Quintino]) of Willmore surfaces into new ones, with a permutability between the two ([Burstall and Quintino]). This paper is dedicated to a self-contained account of the topic in the light-cone picture.