Dressing transformations of constrained Willmore surfaces

We use the dressing method to construct transformations of
constrained Willmore surfaces in arbitrary codimension. An adaptation
of the Terng--Uhlenbeck theory of dressing by simple factors to this
context leads us to define B\"acklund transforms of these surfaces
for which we prove Bianchi permutability. Specialising to
codimension $2$, we generalise the Darboux transforms of Willmore
surfaces via Riccati equations, due to
Burstall--Ferus--Leschke--Pedit--Pinkall, to the constrained Willmore
case and show that they amount to our B\"acklund transforms with real
spectral parameter.