A symmetry-based construction yields rational Lax matrices for the Lambda, H and F families of quadrirational Yang-Baxter maps, with the FIV map left out because it admits no symmetry.
Discrete Lax pairs and hierarchies of integrable difference systems
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abstract
We introduce a family of order $N\in \mathbb{N}$ Lax matrices that is indexed by the natural number $k\in \{1,\ldots,N-1\}.$ For each value of $k$ they serve as strong Lax matrices of a hierarchy of integrable difference systems in edge variables that in turn lead to hierarchies of integrable difference systems in vertex variables or in a combination of edge and vertex variables. Furthermore, the entries of the Lax matrices are considered as elements of a division ring, so we obtain hierarchies of discrete integrable systems extended in the non-commutative domain.
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On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps
A symmetry-based construction yields rational Lax matrices for the Lambda, H and F families of quadrirational Yang-Baxter maps, with the FIV map left out because it admits no symmetry.