Singular self-dual Zollfrei metrics and twistor correspondence

We construct examples of singular self-dual Zollfrei metrics explicitly, by patching a pair of Petean’s self-dual split-signature metrics. We prove that there is a natural one-to-one correspondence between these singular metrics and a certain set of embeddings of RP3RP3 to CP3CP3 which has one singular point. This embedding corresponds to an odd function on RR that is rapidly decreasing and pure imaginary valued. The one-to-one correspondence is explicitly given by using the Radon transform.