  The connection of orthogonal polynomials on the unit circle (OPUC) to the
defocusing Ablowitz-Ladik integrable system involves the definition of a
Poisson structure on the space of Verblunsky coefficients. In this paper, we
compute the complete set of Poisson brackets for the monic orthogonal and the
orthonormal polynomials on the unit circle, as well as for the second kind
polynomials and the Wall polynomials. This answers a question posed by Cantero
and Simon for the case of measures with finite support. We also show that the
results hold for the case of measures with periodic Verblunsky coefficients.
