Finite-dimensional subalgebras in polynomial Lie algebras of rank one

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.