  In this paper we provide a systematic discussion of how to incorporate
orientation preserving symmetries into the treatment of Willmore surfaces via
the loop group method.
  In this context we first develop a general treatment of Willmore surfaces
admitting orientation preserving symmetries, and then show how to induce finite
order rotational symmetries. We also prove, for the symmetric space which is
the target space of the conformal Gauss map of Willmore surfaces in spheres,
the longstanding conjecture of the existence of meromorphic invariant
potentials for the conformal Gauss maps of all compact Willmore surfaces in
spheres. We also illustrate our results by some concrete examples.
