# Abstract integrable systems on hyperk\"ahler manifolds arising from Slodowy slices

## TL;DR

This paper explores holomorphic integrable systems on hyperk"ahler manifolds derived from complex semisimple Lie groups and Slodowy slices, introducing a canonical abstract integrable system and constructing traditional integrable systems with some being completely integrable.

## Contribution

It demonstrates that the hyperk"ahler manifold admits a canonical abstract integrable system and constructs new traditional integrable systems, including some that are completely integrable, based on the argument shift method.

## Key findings

- Existence of a canonical abstract integrable system on the manifold.
- Construction of traditional integrable systems, some fully integrable.
- Application of Mishchenko-Fomenko argument shift approach.

## Abstract

We study holomorphic integrable systems on the hyperk\"ahler manifold $G\times S_{\text{reg}}$, where $G$ is a complex semisimple Lie group and $S_{\text{reg}}$ is the Slodowy slice determined by a regular $\mathfrak{sl}_2(\mathbb{C})$-triple. Our main result is that this manifold carries a canonical \textit{abstract integrable system}, a foliation-theoretic notion recently introduced by Fernandes, Laurent-Gengoux, and Vanhaecke. We also construct traditional integrable systems on $G\times S_{\text{reg}}$, some of which are completely integrable and fundamentally based on Mishchenko and Fomenko's argument shift approach.