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.