  We consider the Szeg\"o kernel for domains \Omega in C^2 given by \Omega =
{(z,w): Im w > b(Re z)} where b is a non-convex quartic polynomial with
positive leading coefficient. Such domains are not pseudoconvex. We describe
the subset of \bar{\Omega} \times \bar{\Omega} on which the kernel and all its
derivatives are finite. In particular, we show that there are points off the
diagonal of the boundary at which the Szeg\"o kernel is infitie as well as
points on the diagonal at which it is finite.
