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

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.