Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation

We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlev\'e equation with $E^{(1)}_6$ symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlev\'e equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlev\'e equation.