On pairs of commuting derivations of the polynomial ring in two variables

Let $k$ be an arbitrary field of characteristic zero, $k[x,y]$ be the polynomial ring and $D$ a $k$-derivation of the ring $k[x,y]$. Recall that a nonconstant polynomial $F\in k[x,y]$ is said to be a Darboux polynomial of the derivation $D$ if $D(F)=\lambda F$ for some polynomial $\lambda \in k[x,y]$. We prove that any two linearly independent over the field $k$ commuting $k$-derivations $D_1$ and $D_2$ of the ring $k[x,y]$ either have a common Darboux polynomial, or $D_1=D_{u_1},D_2=D_{u_2}$ are Jacobian derivations i.e., $D_i(f)=\det J(u_i,f)$ for every $f\in k[x,y],i=1,2,$ where the polynomials $u_1,u_2$ satisfy the condition $\det J(u_{1}, u_{2})=c\in k^{\star}.$ This statement about derivations is an analogue of the known fact from Linear Algebra about common eigenvectors of pairs of commuting linear operators.