Spectral factorization and stochastic realization with zeros on the unit circle

We study the spectral factorization problem of a rational m×m spectral density matrix Φ(z) = Φ(z^-1)^T which has zeros on the unit circle. In the control/estimation literature this situation is either excluded by assuming appropriate stabilizability and detectability conditions or is discussed only partially. Here we provide what seems to be a complete solution, at least in the full-rank case.