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This work is dedicated to the study of the Moebius invariant class ofconstrained Willmore surfaces and its symmetries. We define a spectraldeformation by the action of a loop of flat metric connections; Baecklundtransformations, by applying a dressing action; and, in 4-space, Darbouxtransformations, based on the solution of a Riccati equation. We establish apermutability between spectral deformation and Baecklund transformation andprove that non-trivial Darboux transformation of constrained Willmore surfacesin 4-space can be obtained as a particular case of Baecklund transformation.All these transformations corresponding to the zero multiplier preserve theclass of Willmore surfaces. We verify that, for special choices of parameters,both spectral deformation and Baecklund transformation preserve the class ofconstrained Willmore surfaces admitting a conserved quantity, and, inparticular, the class of CMC surfaces in 3-dimensional space-form.

Constrained Willmore Surfaces: Symmetries of a Moebius Invariant Integrable System