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 $RP^3$ to $CP^3$ which has one singular point. This embedding corresponds to an odd function on $R$ that is rapidly decreasing and pure imaginary valued. The one-to-one correspondence is explicitly given by using the Radon transform.