  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.
