  This paper is a continuation of our study of a class of Toeplitz-like
operators with a rational symbol which has a pole on the unit circle. A
description of the spectrum and its various parts, i.e., point, residual and
continuous spectrum, is given, as well as a description of the essential
spectrum. In this case, the essential spectrum need not be connected in
${\mathbb C}$. Various examples illustrate the results.
