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Orthogonal Laurent polynomials in the unit circle and the theory of Toda-likeintegrable systems are connected using the Gauss--Borel factorization of aCantero-Moral-Velazquez moment matrix, which is constructed in terms of acomplex quasi-definite measure supported in the unit circle. The factorizationof the moment matrix leads to orthogonal Laurent polynomials in the unit circleand the corresponding second kind functions. Jacobi operators, 5-term recursionrelations and Christoffel-Darboux kernels, projecting to particular spaces oftruncated Laurent polynomials, and corresponding Christoffel-Darboux formulaeare obtained within this point of view in a completely algebraic way.Cantero-Moral-Velazquez sequence of Laurent monomials is generalized andrecursion relations, Christoffel-Darboux kernels, projecting to general spacesof truncated Laurent polynomials and corresponding Christoffel-Darboux formulaeare found in this extended context. Continuous deformations of the momentmatrix are introduced and is shown how they induce a time dependantorthogonality problem related to a Toda-type integrable system, which isconnected with the well known Toeplitz lattice. Using the classicalintegrability theory tools the Lax and Zakharov-Shabat equations are obtained.The dynamical system associated with the coefficients of the orthogonal Laurentpolynomials is explicitly derived and compared with the classical Toeplitzlattice dynamical system for the Verblunsky coefficients of Szeg\H{o}polynomials for a positive measure. Discrete flows are introduced and relatedto Darboux transformations. Finally, the representation of the orthogonalLaurent polynomials (and its second kind functions), using the formalism ofMiwa shifts, in terms of $\tau$-functions is presented and bilinear equationsare derived.

Orthogonal Laurent polynomials in unit circle, extended CMV ordering and 2D Toda type integrable hierarchies