  We propose an $n$-dimensional analogue of elliptic difference Painlev\'e
equation. Some Weyl group acts on a family of rational varieties obtained by
successive blow-ups at $m$ points in $\mpp^n(\mc)$, and in many cases they
include the affine Weyl groups with symmetric Cartan matrices as subgroups. It
is shown that the dynamical systems obtained by translations of these affine
Weyl groups possess commuting flows and that their degrees grow quadratically.
For the $E_7^{(1)}$ and $E_8^{(1)}$ cases, existence of preserved quantities is
investigated. The elliptic difference case is also studied.
