# Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation

## TL;DR

This paper classifies integrable modifications of the Ito-Narita-Bogoyavlensky equation using non-invertible discrete transformations, discovering new equations and transformations, including a new discrete equation and restrictions on transformation orders.

## Contribution

It provides the first classification of such modifications in the discrete case, identifying new integrable equations and transformations, and analyzing their nature and limitations.

## Key findings

- Discovered new integrable five-point equations.
- Identified new transformations, including Miura type.
- Derived a new completely discrete equation.

## Abstract

We consider five-point differential-difference equations. Our aim is to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated to transformations of the first, second and third orders. As far as we know, such a classification problem is solved for the first time in the discrete case. We analyze transformations obtained to specify their nature. A number of new integrable five-point equations and new transformations have been found. Moreover, we have derived one new completely discrete equation. There are a few non-standard transformations which are of the Miura type or are linearizable in a non-standard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem.