  We show that an effective action of the one-dimensional torus $\mathbb{G}_m$
on a normal affine algebraic variety $X$ can be extended to an effective action
of a semi-direct product $\mathbb{G}_m\rightthreetimes\mathbb{G}_a$ with the
same general orbit closures if and only if there is a divisor $D$ on $X$ that
consists of $\mathbb{G}_m$-fixed points. This result is applied to the study of
orbits of the automorphism group $\text{Aut}(X)$ on $X$.
