Simple derivations and their images

In the paper, we prove that the derivation $D=y{\mathrm{\partial }}_x+(a_2(x)y^2+a_1(x)y+a_0(x)){\mathrm{\partial }}_y$ of $K[x,y]$ with $a_2(x),a_1(x),a_0(x)\in K[x]$ is simple iff the following conditions hold: $(1)$ $a_0(x)\in K^{\ast }$, $(2)$ $\mathrm{deg}{a}_1(x)\ge 1$ or $\mathrm{deg}{a}_2(x)\ge 1$, $(3)$ there exist no $l\in K^{\ast }$ such that $a_2(x)=l{a}_1(x)-l^2{a}_0(x)$. In addition, we prove that the image of the derivation $D=\partial_x+{\sum_{i=1}^n \gamma_i(x) y_i^{k_i}}{\partial_i}$isaMathieu-Zhaospaceiff$D$ is locally finite. Moreover, we prove that the image of the derivation $D={\sum_{i=1}^n \gamma_i y_i^{k_i}}{\partial_i}$of$K[y_1,\ldots,y_n]$isaMathieu-Zhaospaceiff$k_i\leq 1$forall$1\leq i\leq n$,$n\geq 2$.