On nice ${\mathbb{G}}_m$-actions arising from locally nilpotent derivations with slice

Let $k$ be an algebraically closed field of characteristic zero and $B$ a finitely generated $k$-domain. Given a locally nilpotent derivation $D$ on $B$ admitting a slice $s$, the derivation $\mathrm{\partial }=NsD$ ($N\in \mathbb{Z}\setminus \{0\}$) is semisimple and defines a regular ${\mathbb{G}}_m$-action on $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(B)$. We show that this derivation provides a new explicit description of the ${\mathbb{G}}_m$-action introduced by Freudenburg in terms of the infinitesimal generator $\mathrm{\partial }=NsD$. In the nice case ($D^2(x_i)=0$ for all generators), we prove a linearizability criterion: the associated ${\mathbb{G}}_m$-action is linearizable if and only if $D$ is automorphically conjugate to $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}_n$ and the slice becomes affine-linear in the distinguished variable; moreover, this criterion is independent of the choice of slice.