Planar 2-homogeneous commutative rational vector fields

TL;DR

This paper characterizes commuting 2-homogeneous rational vector fields in two dimensions, showing they can be explicitly integrated into rational or algebraic flows with specific orbit structures, and provides a method to construct such flows.

Contribution

It offers a complete classification of commuting 2-homogeneous rational vector fields and introduces an exhaustive construction method for algebraic flows with high degrees.

Findings

Commuting 2-homogeneous rational vector fields are either rationally integrable or algebraically integrable.

Explicit forms of flows are given in terms of rational functions or algebraic functions.

The construction method can produce flows with arbitrarily high degrees.

Abstract

In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In the latter case, orbits of each flow are given in terms of $1$-homogeneous rational functions $W$ as curves $W(x,y)=\textrm{const}$. An exhaustive method to construct such commuting algebraic flows is presented. The degree of the so-obtained algebraic functions in two variables can be arbitrarily high.