How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6

TL;DR

This paper introduces a new family of integrable quadratic Cremona maps that serve as Kahan discretizations of novel quadratic vector fields with degree 6 polynomial integrals, extending the understanding of discretization of non-integrable systems.

Contribution

It presents a new integrable family of quadratic Cremona maps and a method to repair non-integrable Kahan discretizations through coefficient adjustments.

Findings

Identified a family of integrable quadratic Cremona maps preserving degree 6 elliptic curves.

Demonstrated that straightforward Kahan discretization of certain systems is non-integrable.

Proposed a correction method to restore integrability in Kahan discretizations.

Abstract

We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order $O(\epsilon^2)$ in the coefficients of the discretization, where $\epsilon$ is the stepsize.