Self-dual Zollfrei conformal structures with alpha-surface foliation

A global twistor correspondence is established for neutral self-dual conformal structures with alpha-surface foliation when the structure is close to the standard structure on S^2 times S^2. We need to introduce some singularity for the alpha-surface foliation such that the leaves intersect on a fixed two sphere. In this correspondence, we prove that a natural double fibration is induced on some quotient spaces which is equal to the standard double fibration for the standard Zoll projective structure. We also give a local general forms of neutral self-dual metrics with alpha-surface foliation.