  We start the investigation of immersions $\Psi$ of a simply connected domain
$D$ into three dimensional Euclidean space $R^3$, which have constant mean
curvature (CMC-immersions), and allow for a group of automorphisms of $D$ which
leave the image $\Psi(D)$ invariant. On one hand, this leads to a detailed
description of symmetric CMC-surfaces and the associated symmetry groups. On
the other hand, it allows us to start the classification of CMC-immersions of
an arbitrary, compact or noncompact Riemann surface $M$ into $R^3$ in terms of
Weierstrass-type data, as introduced by Pedit, Wu, and one of the authors [D].
We use our general results to prove, that there are no CMC-tori or Delaunay
surfaces in the dressing orbit of the cylinder. As an example, we apply the
discussion to Smyth surfaces and to a CMC-surface with a branchpoint.

Differential Geometry;Differential Geometry