Self-dual Zollfrei conformal structures with αα-surface foliation

A global twistor correspondence is established for neutral self-dual conformal structures with αα-surface foliation when the structure is close to the standard structure on S2×S2S2×S2. We need to introduce some singularity for the αα-surface foliation such that the leaves intersect on a fixed 2-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 local general forms of neutral self-dual metrics with αα-surface foliation.