Let f be a regular non-constant symbol defined on the d-dimensional torus T-d with values on the unit circle. Denote respectively by K and L, its set of critical points and the associated Laurent operator on l(2)(Z(d)). Let U be a suitable unitary local perturbation of L. We show that the operator U has finite point spectrum and no singular continuous component away from the set f (K). We apply these results and provide a new approach to analyze the spectral properties of GOT matrices with asymptotically constant Verblunsky coefficients. The proofs are based on positive commutator techniques. We also obtain some propagation estimates. (C) 2015 Elsevier Inc. All rights reserved.