Non-Moishezon twistor spaces of with non-trivial automorphism group

We show that a twistor space of a self-dual metric on with -isometry is not Moishezon iff there is a -orbit biholomorphic to a smooth elliptic curve, where the -action is the complexification of the -action on the twistor space. It follows that the -isometry has a two-sphere whose isotropy group is . We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreußler is affirmative even though a twistor space is required to have a non-trivial automorphism group.