  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.
