  We consider the Szeg\"o kernel for non-pseudoconvex domains in C^2 given by
\Omega = {(z,w): Im w > b(Re z)} for b a non-convex even-degree polynomial with
positive leading coefficient. This is an extension of results previously
obtained by the authors for the case in which b has degree 4. We show that the
Szeg\"o kernel has singularities off the diagonal of the boundary of
\bar{\Omega} \times \bar{\Omega} for all such domains, as well as points on the
diagonal of the boundary at which it is finite.
