# On integrability of transverse Lie-Poisson structures at nilpotent elements

## TL;DR

This paper develops a method to construct integrable systems within transverse Lie-Poisson structures at nilpotent elements of simple Lie algebras, using the argument shift technique, and provides a uniform approach for many cases.

## Contribution

It introduces a new uniform construction of polynomial integrable systems for a broad class of nilpotent elements in semisimple Lie algebras.

## Key findings

- Constructed families of functions in involution for transverse Poisson structures.
- Identified completely integrable polynomial systems within these families.
- Provided a systematic method applicable to an infinite class of nilpotent elements.

## Abstract

We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.