Let $\varphi\colon (\mathbb C^\*)^m\times\mathbb A^n\to\mathbb A^n$ be an algebraic effective action of a torus on affine space and $X$ be a Zariski closure of an orbit of this action. Suppose we also have an algebraic action $\psi\colon\mathbb C^\*\times X\to X$ of one-dimensional torus on $X$. Does always exist a morphism $f\colon\mathbb C^\*\to (\mathbb C^\*)^m$ such that $\psi=\varphi\circ f$?



Action of $\mathbb C^*$ on a closed orbit of a torus
Comment above posted as an answer. No, that is not true. One counterexample is when $X=\mathbb{A}^1 = \text{Spec} \ \mathbb{C}[x]$, $m$ equals $1$, $\phi(\lambda,x)$ equals $\lambda x$, and $\psi(\lambda,x)$ equals $\lambda(x-1)+1$.


