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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 ${\rm gl}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2={\rm const}\cdot S$. 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 ${\rm gl}_N$ Landau-Lifshitz model are derived from the Zakharov-Shabat equations. The model is simplified when ${\rm rank}(S)=1$. In this case the Hamiltonian description is suggested. The described family of models includes the elliptic model coming from ${\rm GL}_N$ 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.