  We define certain deformations between minimal and non-minimal constant mean
curvature (CMC) surfaces in Euclidean space $E^3$ which preserve the Hopf
differential. We prove that, given a CMC $H$ surface $f$, either minimal or
not, and a fixed basepoint $z_0$ on this surface, there is a naturally defined
family $f_h$, for all real $h$, of CMC $h$ surfaces that are tangent to $f$ at
$z_0$, and which have the same Hopf differential. Given the classical
Weierstrass data for a minimal surface, we give an explicit formula for the
generalized Weierstrass data for the non-minimal surfaces $f_h$, and vice
versa. As an application, we use this to give a well-defined dressing action on
the class of minimal surfaces. In addition, we show that symmetries of certain
types associated with the basepoint are preserved under the deformation, and
this gives a canonical choice of basepoint for surfaces with symmetries. We use
this to define new examples of non-minimal CMC surfaces naturally associated to
known minimal surfaces with symmetries.
