Magnetic Killing tensors and first integrals of the magnetic flow

TL;DR

This paper introduces magnetic Killing tensors, a new class of symmetric tensors, to construct first integrals for magnetic flows, demonstrating integrability on specific nilmanifolds.

Contribution

The paper defines magnetic Killing tensors and uses them to establish integrability of magnetic flows on certain nilmanifolds, extending classical Killing tensor theory.

Findings

Introduction of magnetic Killing symmetric tensors

Construction of first integrals for magnetic flows

Proven integrability on Kodaira-Thurston and Heisenberg nilmanifolds

Abstract

In this work we introduce a new family of symmetric tensors generalizing Killing tensors, that we call magnetic Killing symmetric tensors. We make use of them to construct first integrals for the magnetic flow associated to a given magnetic field. We apply the results to prove integrability of some invariant magnetic flows (either exact or non-exact) on some 2-step nilmanifolds: the Kodaira-Thurston manifold and Heisenberg nilmanifolds of higher dimensions.