# Multiplicative equations related to the affine Weyl group E$_8$

## TL;DR

This paper derives new integrable and linearisable equations inspired by the affine Weyl group E8, including deautonomised forms related to discrete Painlevé equations and their connections to larger affine Weyl groups.

## Contribution

It introduces five new systems related to E8, identifies their integrability properties, and links some to known Painlevé equations and larger affine Weyl groups.

## Key findings

- Three systems are linearisable with explicit linearisation.
- Two systems are integrable via elliptic functions.
- Deautonomised systems relate to discrete Painlevé equations and larger affine Weyl groups.

## Abstract

We derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated to the affine Weyl group E$_8^{(1)}$. Five such systems are obtained, three of which turn out to be linearisable and the remaining two are integrable in terms of elliptic functions. In the case of the linearisable mappings we derive nonautonomous forms which contain a free function of the dependent variable and we present the linearisation in each case. The two remaining systems are deautonomised to new discrete Painlev\'e equations. We show that these equations are in fact special forms of much richer systems associated to the affine Weyl groups E$_7^{(1)}$ and E$_8^{(1)}$ respectively.