ON NONISOSPECTRAL FLOWS, PAINLEVE EQUATIONS, AND SYMMETRIES OF DIFFERENTIAL AND DIFFERENCE-EQUATIONS

We identify the Painleve Lax pairs with those corresponding to station ary solutions of non-isospectral flows, both for partial differential equations and differential-difference equations. We discuss symmetry r eductions of integrable differential-difference equations and show tha t, in contrast with the continuous case, where Painleve' equations nat urally arise, in the discrete case the so-called ''discrete Painleve e quations'' cannot be obtained in this way. Actually, symmetry reductio ns of integrable differential-difference equations naturally provide ' 'delay Painleve equations. ''