# Lax Pairs for Linear Hamiltonian Systems

## TL;DR

This paper constructs Lax pairs for linear Hamiltonian systems using Gr"obner bases, proving their Poisson nature and exploring properties of associated first integrals, advancing the understanding of integrability in such systems.

## Contribution

It introduces a method to construct Lax pairs for linear Hamiltonian systems and proves their Poisson property, providing new tools for analyzing integrability.

## Key findings

- Lax pairs are explicitly constructed for linear Hamiltonian systems.
- The maps in the Lax pairs are shown to be Poisson.
- Properties of first integrals derived from Lax pairs are investigated.

## Abstract

In the paper Lax pairs for linear Hamiltonian systems of differential equations are constructed. In particular, Gr\"obner bases are used for the computations. It is proved that the maps which appear in the construction of Lax pairs are Poisson. Various properties of first integrals of the system which are obtained from the Lax pairs are investigated.