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In this note, we study the rational covariance extension problem with degree bound when the chosen pseudopolynomial of degree at most n has zeros on the boundary of the unit circle and derive some new theoretical results for this special case. In particular, a necessary and sufficient condition for a solution to be bounded (i.e., has no poles on the unit circle) is established. Our approach is based on convex optimization, similar in spirit to the recent development of a theory of generalized interpolation with a complexity constraint. However, the two treatments do not proceed in the same way and there are important differences between them which we discuss herein. An implication of our results is that bounded solutions can be computed via methods that have been developed for pseudopolynomials which are free of zeros on the boundary, extending the utility of those methods. Numerical examples are provided for illustration.