Lax representations for the three-dimensional Euler--Helmholtz equation

TL;DR

This paper investigates Lax representations for the 3D Euler--Helmholtz equation, demonstrating the non-removability of parameters in existing representations and introducing two new Lax representations with non-removable parameters.

Contribution

It proves the non-removability of parameters in a known Lax representation and presents two novel Lax representations with non-removable parameters.

Findings

Parameter in existing Lax representation is non-removable

Two new Lax representations with non-removable parameters are introduced

Enhances understanding of integrability structures of the Euler--Helmholtz equation

Abstract

The paper is concerned with Lax representations for the three-dimensional Euler--Helmholtz equation. We show that the parameter in the Lax representation from Theorem 3 in [15] is non-removable. Then we present two new Lax representations with non-removable parameters.