# Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries

## TL;DR

This paper investigates the existence of steady ideal magnetofluid solutions lacking continuous Euclidean symmetries, revealing that nontrivial solutions are locally symmetric but can be non-invariant under Euclidean transformations, with explicit examples provided.

## Contribution

It demonstrates the existence of smooth and square integrable magnetofluid solutions without continuous Euclidean symmetries, expanding understanding of symmetry properties in ideal MHD and Euler systems.

## Key findings

- Nontrivial magnetofluidostatic solutions are locally symmetric.
- Existence of force-free and non-force-free solutions without Euclidean invariance.
- Explicit examples of smooth solutions without continuous Euclidean symmetries.

## Abstract

We study the existence of steady solutions of ideal magnetofluid systems (ideal MHD and ideal Euler equations) without continuous Euclidean symmetries. It is shown that all nontrivial magnetofluidostatic solutions are locally symmetric, although the symmetry is not necessarily an Euclidean isometry. Furthermore, magnetofluidostatic equations admit both force-free (Beltrami type) and non-force-free (with finite pressure gradients) solutions that do not exhibit invariance under translations, rotations, or their combination. Examples of smooth solutions without continuous Euclidean symmetries in bounded domains are given. Finally, the existence of square integrable solutions of the tangential boundary value problem without continuous Euclidean symmetries is proved.