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 $f\in K[x]$, where $K$ is a field of characteristic zero, and $d$ the derivation that corresponds to the differential equation $\ddot{x}=f(x)$ in a standard way. Let also $H$ be the Hamiltonian polynomial for $d$, that is $H=\frac{1}{2}{y}^2-\int f(x)dx$. It is known that the set of all polynomial derivations that commute with $d$ forms a $K[H]$-module $M_d$. In this paper, we show that, for every such $d$, the module $M_d$ is of rank $1$ if and only if $\text{deg}f⩾2$. For example, the classical elliptic equation $\ddot{x}=6x^2+a$, where $a\in \mathbb{C}$, falls into this category.