# Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties

## TL;DR

This paper investigates the completeness of Poisson-commutative subalgebras in the symmetric algebra of a semisimple Lie algebra, focusing on their role in integrable systems on coadjoint orbits and flag varieties.

## Contribution

It provides new results on the completeness of Mishchenko-Fomenko and Gelfand-Tsetlin subalgebras in the context of non-regular coadjoint orbits.

## Key findings

- Proves completeness of certain Poisson-commutative subalgebras on coadjoint orbits.
- Establishes connections between integrable systems and flag varieties.
- Extends known results to non-regular orbits.

## Abstract

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $\mathfrak g$, we obtain several results on completeness of homogeneous Poisson-commutative subalgebras of ${\mathcal S}(\mathfrak g)$ on coadjoint orbits. This concerns, in particular, Mishchenko-Fomenko and Gelfand-Tsetlin subalgebras.