Centralizers of Jacobian derivations

Abstract

Let \(\mathbb K\) be an algebraically closed field of characteristic zero, \(\mathbb K[x, y]\) the polynomial ring in variables \(x\), \(y\) and let \(W_2(\mathbb K)\) be the Lie algebra of all \(\mathbb K\)-derivations on \(\mathbb K[x, y]\). A derivation \(D \in W_2(\mathbb K)\) is called a Jacobian derivation if there exists \(f \in \mathbb K[x, y]\) such that \(D(h) = \det J(f, h)\) for any \(h \in \mathbb K[x, y]\) (here \(J(f, h)\) is the Jacobian matrix for \(f\) and \(h\)). Such a derivation is denoted by \(D_f\) . The kernel of \(D_f\) in \(\mathbb K[x, y]\) is a subalgebra \(\mathbb K[p]\) where \(p=p(x, y)\) is a polynomial of smallest degree such that \(f(x, y) = \varphi (p(x, y))\) for some \(\varphi (t) \in \mathbb K[t]\). Let \(C = C_{W_2(\mathbb K)} (D_f)\) be the centralizer of \(D_f\) in \(W_2(\mathbb K)\). We prove that \(C\) is the free \(\mathbb K[p]\)-module of rank \(1\) or \(2\) over \(\mathbb K[p]\) and point out a criterion of being a module of rank \(2\). These results are used to obtain a class of integrable autonomous systems of differential equations.