Reductions on the Lattice and Painleve Equations P2, P5, P6

The symmetry analysis of the discrete integrable systems of the Adler-Bobenko-Suris classification is reviewed, and symmetry reductions of the latter are discussed. In particular, reductions of the discrete potential KdV equation to discrete Painleve equations are presented, and continuous symmetric reductions leading to integrable systems of partial differential equations are considered. As a by-product, an interpretation of solutions of continuous Painleve equations as particular solutions of the discrete potential KdV and a discrete Painleve II equations is demonstrated.