  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 $\mz \times
\wt{W}(A_3^{(1)}) \times \wt{W}(A_1^{(1)})$.
