  We show that generic symplectic quotients of a Hamiltonian $G$-space $M$ by
the action of a compact connected Lie group $G$ are also symplectic quotients
of the same manifold $M$ by a compact torus. The torus action in question
arises from certain integrable systems on $\mathfrak{g}^*$, the dual of the Lie
algebra of $G$. Examples of such integrable systems include the Gelfand-Cetlin
systems of Guillemin-Sternberg in the case of unitary and special orthogonal
groups, and certain integrable systems constructed for all compact connected
Lie groups by Hoffman-Lane. Our abelianization result holds for smooth
quotients, and more generally for quotients which are stratified symplectic
spaces in the sense of Sjamaar-Lerman.
