# Miura transformations for discrete Painlev\'e equations coming from the affine E$_8$ Weyl group

## TL;DR

This paper derives new integrable discrete equations from affine Weyl group symmetries, introduces Miura transformations to prove their integrability, and connects them to known Painlevé equations, expanding understanding of discrete integrable systems.

## Contribution

It presents two new integrable systems from affine E8 Weyl group deautonomisation, with one being linearisable and the other related to affine E7, using Miura transformations for proofs.

## Key findings

- Derived two new integrable discrete systems from affine E8 symmetry.
- Proved one system is a discrete Painlevé equation related to E7.
- Linearised the second system using Miura transformations.

## Abstract

We derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine Weyl group E$_8^{(1)}$. By deautonomisation we obtain two hitherto unknown systems, one of which turns out to be a linearisable one, and we show that both these systems arise from the deautonomisation of a non-QRT mapping. In order to unambiguously prove the integrability of these nonautonomous systems, we introduce a series of Miura transformations which allows us to prove that one of these systems is indeed a discrete Painlev\'e equation, related to the affine Weyl group E$_7^{(1)}$, and to cast it in canonical form. A similar sequence of Miura transformations allows us to effectively linearise the second system we obtain. An interesting off-shoot of our calculations is that the series of Miura transformations, when applied at the autonomous limit, allows one to transform a non-QRT invariant into a QRT one.