Gauge equivalence of $1+1$ Calogero–Moser–Sutherland field theory and a higher-rank trigonometric Landau–Lifshitz model

We consider the classical integrable $(1+1)$ trigonometric $gl_N$ Landau–Lifshitz models constructed by means of quantum $R$-matrices that also satisfy the associative Yang–Baxter equation. It is shown that a $(1+1)$ field analogue of the trigonometric Calogero–Moser–Sutherland model is gauge equivalent to the Landau–Lifshitz model that arises from the Antonov–Hasegawa–Zabrodin trigonometric nonstandard $R$-matrix. The latter generalizes Cherednik's $7$-vertex $R$-matrix in the $GL_2$ case to the case of $GL_N$. An explicit change of variables between the $(1+1)$ models is obtained.