  We define a hierarchy of special classes of constrained Willmore surfaces by
means of the existence of a polynomial conserved quantity of some type,
filtered by an integer. Type 1 with parallel top term characterises parallel
mean curvature surfaces and, in codimension 1, type 1 characterises constant
mean curvature surfaces. We show that this hierarchy is preserved under both
spectral deformation and Baecklund transformation, for special choices of
parameters, defining, in particular, transformations of constant mean curvature
surfaces into new ones, with preservation of the mean curvature, in the latter
case.
