  The modern version of conformal matrix model (CMM) describes conformal blocks
in the Dijkgraaf-Vafa phase. Therefore it possesses a determinant
representation and becomes a Toda chain $\tau$-function only after a peculiar
Fourier transform in internal dimensions. Moreover, in CMM Hirota equations
arise in a peculiar discrete form (when the couplings of CMM are actually Miwa
time-variables). Instead, this integrability property is actually independent
on the measure in the original hypergeometric integral. To get hypergeometric
functions, one needs to pick up a very special $\tau$-function satisfying an
additional "string equation". Usually, its role is played by the lowest
$L_{-1}$ Virasoro constraint, but, in the Miwa variables, it turns into a
finite-difference equation with respect to the Miwa variables. One can get rid
of these differences by rewriting the string equation in terms of some double
ratios of the shifted $\tau$-functions, and then these ratios satisfy more
sophisticated 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 with
non-unit multiplicities, what finally converts the simple discrete Painlev\'e
$q$-PVI into sophisticated differential Painlev\'e VI equations, which will be
considered elsewhere.
