  In the paper, we give an affirmative answer to the conjecture in \cite{13}.
We prove that a Shamsuddin derivation $D$ is simple if and only if
$\operatorname{Aut}(K[x,y_1,\allowbreak\ldots,y_n])_D=\{id\}$. In addition, we
calculate the isotropy groups of the Shamsuddin derivations
$d=\partial_x+\sum_{j=1}^r(a(x)y_j+b_j(x))\partial_j$ of $K[x,y_1,\ldots,y_r]$.
We also prove that $d$ is a Mathieu-Zhao subspace if and only if $a(x)\in K$.
