  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. Jacobi operators, 5-term recursion
relations and Christoffel-Darboux kernels, projecting to particular spaces of
truncated Laurent polynomials, and corresponding Christoffel-Darboux formulae
are obtained within this point of view in a completely algebraic way.
Cantero-Moral-Velazquez sequence of Laurent monomials is generalized and
recursion relations, Christoffel-Darboux kernels, projecting to general spaces
of truncated Laurent polynomials and corresponding Christoffel-Darboux formulae
are found in this extended context. Continuous deformations of the moment
matrix are introduced and is shown how they induce a time dependant
orthogonality problem related to a Toda-type integrable system, which is
connected with the well known Toeplitz lattice. Using the classical
integrability theory tools the Lax and Zakharov-Shabat equations are obtained.
The dynamical system associated with the coefficients of the orthogonal Laurent
polynomials is explicitly derived and compared with the classical Toeplitz
lattice dynamical system for the Verblunsky coefficients of Szeg\H{o}
polynomials for a positive measure. Discrete flows are introduced and related
to Darboux transformations. Finally, the representation of the orthogonal
Laurent polynomials (and its second kind functions), using the formalism of
Miwa shifts, in terms of $\tau$-functions is presented and bilinear equations
are derived.
