# A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations

## TL;DR

This paper investigates a simple scalar integrable PDE related to well-known equations, providing its Lax pair, solutions, symmetries, conservation laws, and reductions, enriching the understanding of its integrability properties.

## Contribution

It introduces a new integrable equation, deriving its Lax pair, solutions, symmetries, conservation laws, and reductions, expanding the class of known integrable systems.

## Key findings

- Derived a Lax pair for the equation
- Constructed soliton and merging soliton solutions
- Established infinite hierarchies of conservation laws and symmetries

## Abstract

We study the simple-looking scalar integrable equation $f_{xxt} - 3(f_x f_t - 1) = 0$, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a B\"acklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlev\'e series, a scaling reduction to a third order ODE and its Painlev\'e series, and the Hirota form (giving further multisoliton solutions).