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The modern version of conformal matrix model (CMM) describes conformal blocksin the Dijkgraaf-Vafa phase. Therefore it possesses a determinantrepresentation and becomes a Toda chain $\tau$-function only after a peculiarFourier transform in internal dimensions. Moreover, in CMM Hirota equationsarise in a peculiar discrete form (when the couplings of CMM are actually Miwatime-variables). Instead, this integrability property is actually independenton the measure in the original hypergeometric integral. To get hypergeometricfunctions, one needs to pick up a very special $\tau$-function satisfying anadditional "string equation". Usually, its role is played by the lowest$L_{-1}$ Virasoro constraint, but, in the Miwa variables, it turns into afinite-difference equation with respect to the Miwa variables. One can get ridof these differences by rewriting the string equation in terms of some doubleratios of the shifted $\tau$-functions, and then these ratios satisfy moresophisticated equations equivalent to the discrete Painlev\'e equations by M.Jimbo and H. Sakai ($q$-PVI equation). They look much simpler in the$q$-deformed ($"5d"$) matrix model, while in the "continuous" limit$q\longrightarrow 1$ to $4d$ one should consider the Miwa variables withnon-unit multiplicities, what finally converts the simple discrete Painlev\'e$q$-PVI into sophisticated differential Painlev\'e VI equations, which will beconsidered elsewhere.

Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models