Action of derivations on polynomials and on Jacobian derivations

Let $\mathbb K$ be a field of characteristic zero, $A := \mathbb K[x_{1}, x_{2}]$ the polynomial ring and $W_2(\mathbb K)$ the Lie algebra of all $\mathbb K$-derivations on $A$. Every polynomial $f \in A$ defines a Jacobian derivation $D_f\in W_2(\mathbb K)$ by the rule $D_f(h)=\det J(f, h)$ for any $h\in A$, where $J(f, h)$ is the Jacobi matrix for $f, h$. The Lie algebra $W_2(\mathbb K)$ acts naturally on $A$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $W_2(\mathbb K)$. It is proved that for a Jordan chain $T(f_1)=\lambda f_1+f_2$, ..., $T(f_{k-1})=\lambda f_{k-1}+f_k$, $T(f_k)=\lambda f_k$ for a derivation $T\in W_2(\mathbb K)$ on $A$ there exists an analogous chain $[T,D_{f_1}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_1} + D_{f_2}$, ..., $[T,D_{f_{k}}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_{k}}$ in $W_2(\mathbb K)$. In case $A:=\mathbb K[x_1, \ldots , x_n]$, the action of normalizers of elements $f$ from $A$ in $W_n(\mathbb K)$ on the principal ideals $(f)$ is considered.