Centralizers of Jacobian derivations

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero, $\mathbb{K}[x,y]$ the polynonial ring in variables $x$, $y$ and let $W_2(\mathbb{K})$ be the Lie algebra of all $\mathbb{K}$-derivations on $\mathbb{K}[x,y]$. A derivation $D\in W_2(\mathbb{K})$ is called a Jacobian derivation if there exists $f\in \mathbb{K}[x,y]$ such that $D(h)=\det J(f,h)$ for any $h\in \mathbb{K}[x,y]$ (here $J(f,h)$ is the Jacobian matrix for $f$ and $h$). Such a derivation is denoted by $D_f$. The kernel of $D_f$ in $\mathbb{K}[x,y]$ is a subalgebra $\mathbb{K}[p]$ where $p=p(x,y)$ is a polynomial of smallest degree such that $f(x,y)=\phi (p(x,y)$ for some $\phi (t)\in \mathbb{K}[t]$. Let $C=C_{W_2(\mathbb{K})}(D_f)$ be the centralizer of $D_f$ in $W_2(\mathbb{K})$. We prove that $C$ is the free $\mathbb{K}[p]$-module of rank 1 or 2 over $\mathbb{K}[p]$ and point out a criterion of being a module of rank $2$. These results are used to obtain a class of integrable autonomous systems of differential equations.