On non-isospectral flows, Painlevé equations, and symmetries of differential and difference equations

We identify the Painlevé Lax pairs with those corresponding to stationary solutions of non-isospectral flows, both for partial differential equations and differential-difference equations. We discuss symmetry reductions of integrable differential-difference equations and show that, in contrast with the continuous case, where Painlevé equations naturally arise, in the discrete case the so-called “discrete Painlevé equations” cannot be obtained in this way. Actually, symmetry reductions of integrable differential-difference equations naturally provide “delay Painlevé equations”.