Centralizers of Jacobian derivations

Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.