  Let k be an algebraically closed field of zero characteristic. The Lie
algebra W_2 of all k-derivations of the polynomial ring k[x, y] naturally acts
on the polynomial ring k[x, y] and also on the field of rational functions k(x,
y). For a fixed non-constant rational function u from k(x,y) we consider the
set A_{W_2}(u) of all derivations D from W_2 such that D(u)=0. We prove that
A_{W_2}(u) is a free submodule of rank 1 of the k[x,y]-module W_2. A
description of the maximal abelian subalgebras as well of the centralizers of
elements in the Lie algebra A_{W_2}(u) has been obtained.
