Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential

We define certain deformations between minimal and non-minimal constant mean curvature (CMC) surfaces in Euclidean space E3 which preserve the Hopf differential. We prove that, given a CMC H surface f, either minimal or not, and a fixed basepoint z0 on this surface, there is a naturally defined family fh, for all real h, of CMC h surfaces that are tangent to f at z0, 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 fh, 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.