# Constructing discrete Painlev\'e equations: from E$_8^{(1)}$ to A$_1^{(1)}$ and back

## TL;DR

This paper introduces and applies the restoration method to systematically derive discrete Painlevé equations, including challenging cases, by starting from known equations and constructing related mappings through homographic transformations.

## Contribution

The paper presents the restoration method for deriving discrete Painlevé equations from known forms, extending its applicability to complex cases without QRT mappings and multiple-step evolutions.

## Key findings

- Successfully derived discrete Painlevé equations from E8^{(1)} symmetry.
- Extended the method to cases lacking QRT mappings.
- Demonstrated multiple-step evolution forms.

## Abstract

The `restoration method' is a novel method we recently introduced for systematically deriving discrete Painlev\'e equations. In this method we start from a given Painlev\'e equation, typically with E$_8^{(1)}$ symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that are equivalent to it by homographic transformations. Discrete Painlev\'e equations are then obtained by deautonomising the various mappings thus obtained. We apply the restoration method to two challenging examples, one of which does not lead to a QRT mapping at the autonomous limit but we verify that even in that case our method is indeed still applicable. For one of the equations we derive we also show how, starting from a form where the independent variable advances one step at a time, we can obtain versions that correspond to multiple-step evolutions.