Involutions of Halphen Pencils of Index 2 and Discrete Integrable Systems

TL;DR

This paper constructs involutions for a specific Halphen pencil of index 2 and demonstrates that the associated birational mapping of the elliptic Painlevé equation can be expressed as their composition, linking geometric involutions to integrable systems.

Contribution

It introduces a novel construction of involutions for Halphen pencils of index 2 and connects these to the birational mappings of elliptic Painlevé equations.

Findings

Involutions for Halphen pencils of index 2 are explicitly constructed.

The birational mapping of the elliptic Painlevé equation is shown to be a composition of two involutions.

The work bridges geometric involutions with discrete integrable systems.

Abstract

We constructed involutions for a Halphen pencil of index 2, and proved that the birational mapping corresponding to the autonomous reduction of the elliptic Painlev\'e equation for the same pencil can be obtained as the composition of two such involutions.