Jordan properties of automorphism groups of certain open algebraic varieties

Let $W$ be a quasiprojective variety over an algebraically closed field of characteristic zero. Assume that $W$ is birational to a product of a smooth projective variety $A$ and the projective line. We prove that if $A$ contains no rational curves then the automorphism group $G:=Aut(W)$ of $W$ is Jordan. That means that there is a positive integer $J=J(W)$ such that every finite subgroup $\mathcal{B}$ of $G$ contains a commutative subgroup $\mathcal{A}$ such that $\mathcal{A}$ is normal in $\mathcal{B}$ and the index $[\mathcal{B}:\mathcal{A}]\le J$ .