Integrable deformations of principal chiral model fromsolutions of associative Yang-Baxter equation

We describe deformations of the classical principal chiral model and the $(1+1)$-dimensional Gaudin model related to the Lie group GLN. 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.