R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\mathrm{B}\mathrm{C}}_n$ type

We consider the elliptic Calogero-Inozemtsev system of ${\mathrm{B}\mathrm{C}}_n$ type with five arbitrary constants and propose $R$-matrix valued generalization for $2n\times 2n$ Takasaki’s Lax pair. For this purpose we extend the Kirillov’s $\mathrm{B}$-type associative Yang-Baxter equations to the similar relations depending on the spectral parameters and the Planck constants. General construction uses the elliptic Shibukawa-Ueno $R$-operator and the Komori-Hikami $K$-operators satisfying reflection equation. Then, using the Felder-Pasquier construction the answer for the Lax pair is also written in terms of the Baxter’s 8-vertex $R$-matrix. As a by-product of the constructed Lax pair we also propose ${\mathrm{B}\mathrm{C}}_n$ type generalization for the elliptic XYZ long-range spin chain, and we present arguments pointing to its integrability.