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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 
$G$, arise naturally in hyperkähler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices’, Math. Res. Let., to appear] use a Hamiltonian 
$G$-action to endow 
$G\times S_{\!\text{reg}}$ with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian 
$G$-actions, 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 
$X\cong G\times S_{\!\text{reg}}$.
