The dressing method for the generating of new solutions of integrable systems was introduced in 1979 by Zakharov and Shabat [16]. It was soon traced back to a natural loop group action on the solution space of integrable systems [1,11]. With the introduction of the theory of integrable systems into geometry, most notably the theory of surfaces of constant mean curvature (CMC surfaces) [10,2], the dressing method also entered the realm of differential geometry.Let it serve as an illustration of the power of the dressing action that, due to [8] (see also [3]), all CMC immersions of finite type, in particular all CMC tori, are included in the dressing orbit of the standard cylinder. This result was used in [4] to reproduce the classification of CMC tori given in [10] in terms of the dressing action.Moreover, while the integrable systems methods in general only apply to a certain class of CMC surfaces, those without umbilics, the dressing action can easily be applied also to CMC surfaces with umbilics. The latter was made possible by a general loop group theoretic approach to CMC surfaces, the so called DPW method [7].Similar to the Weierstraß representation of minimal surfaces, the DPW method starts with a holomorphic function E and a meromorphic function f , both defined on some open, simply connected subset D of the complex plane, and constructs an S 1 -family of isometric conformal CMC immersions Ψ λ∈S 1 : D → R 3 , the associated family, from these data. As a first application of the dressing action, in [6] and [15] it was shown that the dressing action can be used to describe the set of admissible input data (E, f ) for the DPW method.While Edz 2 is simply the Hopf differential of the resulting CMC immersion, the function f has no such simple geometric interpretation. The problem shows especially if one wants to construct CMC immersions Ψ which are invariant under a symmetry group Γ ⊂ AutD of biholomorphic automorphisms of D, i.e. Ψ • γ = T Ψ, where γ ∈ Γ and T is a (proper) Euclidean motion in space. It is clear, that Edz 2 has to be automorphic w.r.t. Γ, i.e. γ * (Edz 2 ) = Edz 2 or E • γ = (γ ′ ) −2 E. But the meromorphic function f satisfies no such automorphicity conditions. Instead, as was shown by the authors in [5], f transforms by complicated dressing transformations under Γ. For further reference see [5]. Thus, in the DPW formulation the understanding of compact and symmetric CMC immersions is intimately related to the understanding of the dressing action. This served as a strong incentive to further investigate the dressing action, in particular on the meromorphic data of the DPW method.Clearly, the Hopf differential is invariant under dressing. Wu [14] constructed a large set of algebraic invariants under dressing and was able to find normalized representatives in each dressing orbit [13]. These results are particularly useful, since the dressing orbits, as the finite type results indicate, are very large.