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

Geometry of the space of sections of twistor spaces with circle action