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As regard to their inherent severe drawbacks (namely ill-conditioning and sometimes over-modeling problems), the blind second-order equalization schemes need refinements for them to be tractable in realistic contexts. We propose to force cyclo-stationarity at the emission: this statistical property remains for the observation process and gives rise to a spectral factorization problem exhibiting a symmetry. This is solved via a structured subspace method yielding a unique solution. In other words, the symmetry implies that the identification of the unknown channel (supposed to be causal FIR and scalar-valued, i.e. single sensor/sampling at the symbol period) may be achieved consistently-whenever the order is over-determined and without any constraint on the position of the zeros of this channel.