We study the birational representation of $\wt{ W}(A_1^{ (1)}\times A_3^{ (1)})$ proposed by Kajiwara-Noumi-Yamada (KNY) in the case of $m=2$ and $n=4$. It is shown that the equation can be lifted to an automorphism of a family of $A_3^{ (1)}$ surfaces and therefore the group of Cremona isometries is $\wt{ W}(D_5^{ (1)})$ ($\supset \wt{ W}(A_1^{ (1)}\times A_3^{ (1)})$). The equation can be decomposed into two mappings which are conjugate to the $q$-$P_{ VI}$ equation. It is also shown that the subgroup of Cremona isometries which commute with the original translation is isomorphic to $\mz \times \wt{ W}(A_3^{ (1)}) \times \wt{ W}(A_1^{ (1)})$.
Categories: nlin.SI