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We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coordinates in a pseudoconcave neighbourhood of anO⁢(1)\mathcal{O}(1)rational curve; in the construction, we present every such neighbourhood asQD/FQ_{\mathbb{D}}/\mathcal{F}for some holomorphic foliation ℱ, whereQDQ_{\mathbb{D}}is an open neighbourhood in the standard quadricQ⊂P2×P2Q\subset\mathbb{P}^{2}\times\mathbb{P}^{2}. As a consequence of the normal coordinates, we obtain a new normal form for Cauchy Riemann structures on the three-sphere that are isotopic to the standard one. We end the paper with explicit calculations for the cases arising from deformations of the normal isolated singularitiesX⁢Y=ZnXY=Z^{n}.

A twistor transform and normal forms for Cauchy Riemann structures