Varieties of the form G×Sreg, where G is a complex semisimple group and Sreg 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×Sreg 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≅G×Sreg.