  Let W_n(K) be the Lie algebra of derivations of the polynomial algebra
K[X]:=K[x_1,...,x_n] over an algebraically closed field K of characteristic
zero. A subalgebra L of W_n(K) is called polynomial if it is a submodule of the
K[X]-module W_n(K). We prove that the centralizer of every nonzero element in L
is abelian provided L has rank one. This allows to classify finite-dimensional
subalgebras in polynomial Lie algebras of rank one.
