On Shamsuddin derivations and the isotropy groups

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 $\mathrm{Aut}(K[x,y_1,\dots ,y_n]{)}_D=\{id\}$. In addition, we calculate the isotropy groups of the Shamsuddin derivations $d={\mathrm{\partial }}_x+{\sum }_{j=1}^r(a(x)y_j+b_j(x)){\mathrm{\partial }}_j$ of $K[x,y_1,\dots ,y_r]$. We also prove that $d$ is a Mathieu-Zhao subspace if and only if $a(x)\in K$.