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We show that, for the unitary component in the Wold-Kolmogorov decomposition of the cogenerator of the semigroup $(\\tilde\\tau_t)_{t\\ge 0}$, {\\it any singular} spectral type may be achieved by $\\gS_1$ perturbations. We provide an explicit construction for a perturbation with a given spectral type based on the theory of model spaces of the Hardy space $H^2$. 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