# The symmetry approach to integrability: recent advances

## TL;DR

This paper reviews recent advances in the symmetry approach to integrability, covering evolution equations, recursion operators, Hamiltonian structures, and non-abelian systems, highlighting new methods for analyzing complex integrable systems.

## Contribution

It introduces new techniques for studying non-diagonalisable systems and non-abelian integrable equations within the symmetry framework.

## Key findings

- Development of formal recursion operators for non-diagonalisable systems
- Extension of symmetry methods to matrix and non-abelian equations
- Enhanced understanding of Hamiltonian and quasi-local operators

## Abstract

We provide a concise introduction to the symmetry approach to integrability. Some results on integrable evolution and systems of evolution equations are reviewed. Quasi-local recursion and Hamiltonian operators are discussed. We further describe non-abelian integrable equations, especially matrix (ODE and PDE) systems. Some non-evolutionary integrable equations are studied using a formulation of formal recursion operators that allows to study non-diagonalisable systems of evolution equations.