  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
$[\mathcal{B}:\mathcal{A}] \le J$ .
