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Yang-Baxter Maps from 3x3 Lax Matrix Reductions in Integrable Systems

A multi-parameter family of Poisson Yang-Baxter maps is built from generic 3x3 binomial Lax matrices L=X - lambda*Ka by solving the Lax refactorisation problem, extended with additional essential parameters. Students learn how low-dimensional reductions on the Casimir level sets of the Sklyanin (r-matrix) Poisson bracket, together with degenerate limits in the parameters, produce new birational Yang-Baxter maps that generalise the Adler-Yamilov map vectorially and remain Liouville integrable.