Action of derivations on polynomials and on Jacobian derivations

Let K be a field of characteristic zero, A := K[x₁, x₂] the polynomial ring and W₂( K) the Lie algebra of all K-derivations on A. Every polynomial f ∈ A defines a Jacobian derivation Df∈ W₂( K) by the rule Df(h)= J(f, h) for any h∈ A, where $J(f, h)$ is the Jacobi matrix for $f, h$. The Lie algebra W₂( 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 W₂( K). It is proved that for a Jordan chain T(f₁)=λ f₁+f₂, ..., T(fₖ₋₁)=λ fₖ₋₁+fₖ, T(fₖ)=λ fₖ for a derivation T∈ W₂( K) on A there exists an analogous chain [T,Df₁]=(λ -div T)Df₁ + Df₂, ..., [T,D_fₖ]=(λ -div T)D_fₖ in W₂( K). In case A:= K[x₁, … , xₙ], the action of normalizers of elements f from A in Wₙ( K) on the principal ideals $(f)$ is considered.