Integrable deformations of principal chiral model from solutions of associative Yang-Baxter equation

We describe deformations of the classical principle chiral model and 1+1 Gaudin model related to ${\mathrm{G}\mathrm{L}}_N$ Lie group. The deformations are generated by $R$-matrices satisfying the associative Yang-Baxter equation. Using the coefficients of the expansion for these $R$-matrices we derive equations of motion based on a certain ansatz for $U$-$V$ pair satisfying the Zakharov-Shabat equation. Another deformation comes from the twist function, which we identify with the cocentral charge in the affine Higgs bundle underlying the Hitchin approach to 2d integrable models.