Let $k$ be a field with $2 \in k^{\times}$. Set $$A:= k[x,y,z]/(y^{2}z = x^{3}+x^{2}z)$$ and let $X := \operatorname{Proj} A$ be the nodal curve. The normalization of $X$ is a morphism $\pi : \mathbb{P}_{k}^{1} \to X$, given by taking Proj of the (injective) degree 3 morphism $A \to B := k[s,t]$ sending $(x,y,z) \mapsto (s(t^{2}-s^{2}),t(t^{2}-s^{2}),s^{3}).$ The node $x_{0} \in X$ is the complement of the open subset $U := D_{+}((x-y)(x+y)) \subset X$ and the restriction $\pi^{-1}(U) \to U$ is an isomorphism; here $\pi^{-1}(U) \simeq D_{+}(-(s-t)^{3}(s+t)^{3}) \subset \mathbb{P}_{k}^{1}$, which is isomorphic to $\mathbb{G}_{m}$ since $2 \in k^{\times}$. Thus there is an open immersion $j : \mathbb{G}_{m} \to X$ whose complement is the node.
Does there exist a $\mathbb{G}_{m}$-action on $X$ such that $j$ is $\mathbb{G}_{m}$-equivariant with respect to the usual action of $\mathbb{G}_{m}$ on itself?
Keywords: nodal cubic, toric variety