How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

Abstract

<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\epsilon ^2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in the coefficients of the discretization, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\epsilon $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is the stepsize.</jats:p>