A Class of simple derivations of polynomial ring $k[x_1,x_2,\dots ,x_n]$

Let $k$ be a field of characteristic zero. Let $m$ and $\alpha$ be positive integers. For $n\ge 2$, let $R_n=k[x_1,x_2,\dots ,x_n]$ with the $k$-derivation $d_n$ given by $d_n=(1-x_1{x}_2^{\alpha }){\mathrm{\partial }}_{x_1}+x_1^m{\mathrm{\partial }}_{x_2}+x_2{\mathrm{\partial }}_{x_3}+\cdots +x_{n-1}{\mathrm{\partial }}_{x_n}$. We prove that for integers $m\ge 2$ and $\alpha \ge 1$, $d_n$ is a simple derivation on $R_n$ and $d_n(R_n)$ contains no units. This generalizes a result of D. A. Jordan. We also show that the isotropy group of $d_n$ is conjugate to a subgroup of translations.