Neutral components of automorphism groups of (semi)rigid affine varieties

We say that an affine variety $X$ is (semi)rigid if all nontrivial actions of the additive group on $X$ have the same invariant ring. Then all such actions comprise the abelian subgroup denoted $\mathrm{SAut}(X)$. We prove that $X$ is (semi)rigid precisely when the neutral component $\mathrm{Aut}^\circ(X)$ of its automorphism group is nested. In this case, the neutral component is the semidirect product of any maximal algebraic torus and $\mathrm{SAut}(X)$. The proof uses Laurent expansions of algebraic curves in $\mathrm{Aut}^\circ(X)$.