Geometry of an elliptic-difference equation related to Q4

In this paper, we investigate a nonlinear non-autonomous elliptic difference equation, which was constructed by Ramani, Carstea and Grammaticos by integrable deautonomization of a periodic reduction of the discrete Krichever-Novikov equation, or Q4. We show how to construct it as a birational mapping on a rational surface blown up at eight points in $\mathbb P^1\times \mathbb P^1$, and find its affine Weyl symmetry, placing it in the geometric framework of the Painlev\'e equations. The initial value space is ell-$A_0^{(1)}$ and its symmetry group is $W(F_4^{(1)})$. We show that the deautonomization is consistent with the lattice-geometry of Q4 by giving an alternative construction, which is a reduction from Q4 in the usual sense. A more symmetric reduction of the same kind provides another example of a second-order integrable elliptic difference equation.