The projective translation equation and rational plane flows. II. Corrections and additions

TL;DR

This paper corrects and extends previous work on classifying algebraic and rational projective flows, providing new methods, classifications, and examples, including symmetric and solenoidal flows, with improved clarity and completeness.

Contribution

It corrects earlier proofs, introduces an inductive method for constructing algebraic flows, and classifies new types of rational flows with specific symmetries and properties.

Findings

Corrected proof of main theorem in prior work.

Developed an inductive method for algebraic projective flows.

Classified all rational flows symmetric under a specific nonlinear involution.

Abstract

In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments and additions are needed to claim the validity of the classification results. With these new results in a disposition, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows. Further, we expand on results concerning rational flows in dimension $2$. Previously we found such flows symmetric with respect to a linear involution $i(x,y)=(y,x)$. Here we find all rational flows symmetric with respect to a non-linear $1$-homogeneous involution $i(x,y)=(\frac{y^2}{x},y)$. We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples.