# A strongly coupled extended Toda hierarchy and its Virasoro symmetry

## TL;DR

This paper introduces a strongly coupled extended Toda hierarchy (SCETH), demonstrating its Virasoro symmetry and providing multi-fold Darboux transformations, thus advancing the understanding of integrable systems and their symmetries.

## Contribution

It constructs the SCETH from a commutative subalgebra of gl(2,C), proving its Virasoro symmetry and deriving Darboux transformations, extending the theory of integrable hierarchies.

## Key findings

- SCETH possesses Virasoro type additional symmetry.
- Multi-fold Darboux transformations for SCETH are derived.
- SCETH generalizes the extended Toda hierarchy and relates to the extended multicomponent Toda hierarchy.

## Abstract

As a generalization of the integrable extended Toda hierarchy and a reduction of the extended multicomponent Toda hierarchy, from the point of a commutative subalgebra of $gl(2,\mathbb{C})$, we construct a strongly coupled extended Toda hierarchy(SCETH) which will be proved to possess a Virasoro type additional symmetry by acting on its tau-function. Further we give the multi-fold Darboux transformations of the strongly coupled extended Toda hierarchy.