Commuting planar polynomial vector fields for conservative Newton systems

Abstract

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let [Formula: see text], where [Formula: see text] is a field of characteristic zero, and [Formula: see text] the derivation that corresponds to the differential equation [Formula: see text] in a standard way. Let also [Formula: see text] be the Hamiltonian polynomial for [Formula: see text], that is [Formula: see text]. It is known that the set of all polynomial derivations that commute with [Formula: see text] forms a [Formula: see text]-module [Formula: see text]. In this paper, we show that, for every such [Formula: see text], the module [Formula: see text] is of rank [Formula: see text] if and only if [Formula: see text]. For example, the classical elliptic equation [Formula: see text], where [Formula: see text], falls into this category.