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 onedimensional torus on $X$. Does always exist a morphism $f\colon\mathbb C^*\to (\mathbb C^*)^m$ such that $\psi=\varphi\circ f$?

1$\begingroup$ No. Consider the "standard" action of $\mathbb{C}^*$ on $\mathbb{A}^1$ by $\lambda\cdot x= \lambda x$, and also the second action of $\mathbb{C}^*$ on $\mathbb{A}^1$ by $\lambda \ast x = \lambda(x1)+1$. $\endgroup$ – Jason Starr Mar 1 '16 at 14:29

$\begingroup$ @JasonStarr Is your second action really an action? It seems to me that $(\lambda \cdot \mu) \ast x \neq \lambda \ast (\mu\ast x)$ $\endgroup$ – Ariyan Javanpeykar Mar 1 '16 at 15:12

$\begingroup$ @AriyanJavanpeykar. Define $y=\lambda(x1)+1$. Then $\mu\ast y$ equals $\mu(y1)+1$. But $y1$ equals $\lambda(x1)$. Thus $\mu\ast y$ equals $\mu\lambda(x1)+1$. So $\mu\ast(\lambda\ast x)$ equals $(\mu\lambda)\ast x$. smiley face $\endgroup$ – Jason Starr Mar 1 '16 at 16:11

$\begingroup$ @JasonStarr Thank you for the explanation. Shame on me. smiley face $\endgroup$ – Ariyan Javanpeykar Mar 1 '16 at 19:00
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 $\lambda(x1)+1$.
In fact $X$ is a toric variety. "In general" the automorphism group of this variety is the torus himself. But in some cases it is not. For instance if $\varphi$ is the diagonal action of $\bf {C^*} ^n$ on $\bf C^n$ the closure of the orbit is the affine space itself and the automorphims group of this space is much bigger ; it contains the homothety group which fixes some point as remarked by Jason Starr, or in higher dimension the entire group $GL(n,\bf C)$ hence any conjugate subrgoup of the diagonal torus, and even more complicated stuff. The complete classification of toric varietes and their automorphism groups is in Demazure, Michel Sousgroupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. (4) 3 1970 507–588.