Third Order Integrable Equation Possessing Symplectic Operator Of Degree 9

TL;DR

This paper classifies third order integrable evolution equations, discovers a new supersymmetric integrable equation with a seventh-order symmetry, and provides its Lax representation and bi-Hamiltonian structures.

Contribution

It introduces a new supersymmetric integrable equation with a symplectic operator of degree 9, expanding the classification of such systems.

Findings

Discovered a new supersymmetric integrable equation.

Provided Lax representation for the new equation.

Established bi-Hamiltonian structures for the equation.

Abstract

We perform a classification of third order integrable systems of evolution equations with respect to higher symmetries. Applying it, we consider polynomial systems that are 0-homogeneous under a suitable weighting of variables with main matrix $***$ having a seventh-order symmetry. A new integrable equation is discovered. For this new supersymmetric equation the Lax representation and bi-Hamiltonian structures are given.