  We study the holomorphic symplectic geometry of (the smooth locus of) the
space of holomorphic sections of a twistor space with rotating circle action.
The twistor space has a meromorphic connection constructed by Hitchin. We give
an interpretation of Hitchin's meromorphic connection in the context of the
Atiyah--Ward transform of the corresponding hyperholomorphic line bundle.
  It is shown that the residue of the meromorphic connection serves as a moment
map for the induced circle action, and its critical points are studied.
Particular emphasis is given to the example of Deligne--Hitchin moduli spaces.
