  Over a field $k$, we study rational UFDs of finite transcendence degree $n$
over $k$. We classify such UFDs $B$ when $n=2$, $k$ is algebraically closed,
and $B$ admits a positive $\mathbb{Z}$-grading, showing in particular that $B$
is affine over $k$. We also consider the Russell cubic threefold over
$\mathbb{C}$, and the Asanuma threefolds over a field of positive
characterstic, showing that these threefolds admit no elliptic
$\mathbb{G}_m$-action. Finally, we show that, if $X$ is an affine $k$-variety
and $X\times\mathbb{A}^m_k\cong_k\mathbb{A}^{n+m}_k$, then
$X\cong_k\mathbb{A}^n_k$ if and only if $X$ admits an elliptic
$\mathbb{G}_m$-action.
