  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.
