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## Higher Rank 1 + 1 Integrable Landau–Lifshitz Field Theories From Associative Yang–Baxter Equation

We propose a construction of 1 + 1 integrable Heisenberg–Landau–Lifshitz type equations in the gl*N* case. The dynamical variables are matrix elements of *N* × *N* matrix *S* with the property *S*2 = const · *S*. The Lax pair with spectral parameter is constructed by means of a quantum *R*-matrix satisfying the associative Yang–Baxter equation. Equations of motion for gl*N* Landau–Lifshitz model are derived from the Zakharov–Shabat equations. The model is simplified when rank(*S*) = 1. In this case the Hamiltonian description is suggested. The described family of models includes the elliptic model coming from GL*N* Baxter–Belavin elliptic *R*‑matrix. In *N* = 2 case the widely known Sklyanin’s elliptic Lax pair for XYZ Landau–Lifshitz equation is reproduced. Our construction is also valid for trigonometric and rational degenerations of the elliptic *R*‑matrix.