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Painlev\`e equation for conformal blocks is a combined corollary ofintegrability and Ward identities, which can be explicitly revealed in thematrix model realization of AGT relations. We demonstrate this in some detail,both for $q$-Painlev\`e equations for the $q$-Virasoro conformal block, or AGTdual gauge theory in $5d$, and for ordinary Painlev\`e equations, or AGT dualgauge theory in $4d$. Especially interesting is the continuous limit from $5d$to $4d$ and its description at the level of equations for eight$\tau$-functions. Half of these equations are governed by integrability andanother half by Ward identities.

AGT correspondence, (q-)Painlev\`e equations and matrix models