Orthogonal Laurent polynomials in the unit circle and the theory of Toda-like
integrable systems are connected using the Gauss--Borel factorization of a
Cantero-Moral-Velazquez moment matrix, which is constructed in terms of a
complex quasi-definite measure supported in the unit circle. The factorization
of the moment matrix leads to orthogonal Laurent polynomials in the unit circle
and the corresponding second kind functions.