AGT correspondence, (q-)Painlevè equations and matrix models

Painlevè equation for conformal blocks is a combined corollary of integrability and Ward identities, which can be explicitly revealed in the matrix model realization AGT relations. We demonstrate this some detail, both q-Painlevè equations q-Virasoro block, or dual gauge theory 5d, ordinary equations, 4d. Especially interesting continuous limit from 5d to 4d its description at level eight τ-functions. Half these are governed by another half identities.