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

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.

15 pages, 4 figures

References in corpus (3)

• Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems
• Manin involutions for elliptic pencils and discrete integrable systems
• On the singularity structure of Kahan discretizations of a class of quadratic vector fields

Cited by in corpus (1)

• A three-dimensional generalization of QRT maps