Reductions and degenerate limits of Yang-Baxter maps with 3 × 3 Lax matrices

We generalise a family of quadrirational parametric Yang-Baxter (YB) maps with 3 × 3 Lax matrices by introducing additional essential parameters. These maps preserve a prescribed Poisson structure which originates from the Sklyanin bracket. We investigate various low-dimensional reductions of this family, as well as degenerate limits with respect to the parameters that were introduced. As a result, we derive several birational YB maps, and we discuss some of their integrability properties. This work is part of a more general classification of YB maps admitting a strong 3 × 3 Lax matrix with a linear dependence on the spectral parameter.