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The quantum elliptic $R$-matrices of Baxter-Belavin type satisfy theassociative Yang-Baxter equation in ${\rm Mat}(N,\mathbb C)^{\otimes 3}$. Thelatter can be considered as noncommutative analogue of the Fay identity for thescalar Kronecker function. In this paper we extend the list of $R$-matrixvalued analogues of elliptic function identities. In particular, we proposecounterparts of the Fay identities in ${\rm Mat}(N,\mathbb C)^{\otimes 2}$. Asan application we construct $R$-matrix valued $2N^2\times 2N^2$ Lax pairs forthe Painlev\'e VI equation (in elliptic form) with four free constants using${\mathbb Z}_N\times {\mathbb Z}_N$ elliptic $R$-matrix. More precisely, thefour free constants case appears for an odd $N$ while even $N$'s correspond toa single constant.

Quantum Baxter-Belavin R-matrices and multidimensional Lax pairs for Painleve VI