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We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard(KZB) equations with $\tilde N$ punctures by deformation of the correspondingquantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limitof the first one brings the model to the ordinary rational KZ equation. Anotherone is $\tau$. At the level of classical mechanics the deformation parameter$\tau$ allows to extend the previously obtained modified Gaudin models to themodified Schlesinger systems. Next, we notice that the identities underlyinggeneric (elliptic) KZB equations follow from some additional relations for theproperly normalized $R$-matrices. The relations are noncommutative analogues ofidentities for (scalar) elliptic functions. The simplest one is the unitaritycondition. The quadratic (in $R$ matrices) relations are generated bynoncommutative Fay identities. In particular, one can derive the quantumYang-Baxter equations from the Fay identities. The cubic relations provideidentities for the KZB equations as well as quadratic relations for theclassical $r$-matrices which can be halves of the classical Yang-Baxterequation. At last we discuss the $R$-matrix valued linear problems whichprovide ${\rm gl}_{\tilde N}$ Calogero-Moser (CM) models and Painleve equationsvia the above mentioned identities. The role of the spectral parameter playsthe Planck constant of the quantum $R$-matrix. When the quantum ${\rm gl}_N$$R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever'sansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tildeN}$ CM models. The linear problems for the quantum CM models generalize the KZequations in the same way as the Lax pairs with spectral parameter generalizethose without it.

Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations