  This short note considers varieties of the form $G\times S_{\text{reg}}$,
where $G$ is a complex semisimple group and $S_{\text{reg}}$ is a regular
Slodowy slice in the Lie algebra of $G$. Such varieties arise naturally in
hyperk\"ahler geometry, theoretical physics, and in the theory of abstract
integrable systems developed by Fernandes, Laurent-Gengoux, and Vanhaecke. In
particular, previous work of the author and Rayan uses a Hamiltonian $G$-action
to endow $G\times S_{\text{reg}}$ with a canonical abstract integrable system.
One might therefore wish to understand, in some sense, all examples of abstract
integrable systems arising from Hamiltonian $G$-actions. Accordingly, we
consider a holomorphic symplectic variety $X$ carrying an abstract integrable
system induced by a Hamiltonian $G$-action. Under certain hypotheses, we show
that there must exist a $G$-equivariant variety isomorphism $X\cong G\times
S_{\text{reg}}$.
