  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.
