A Note on the Automorphism Group of the Bielawski–Pidstrygach Quiver

We show that there exists a morphism between a group $\Gamma^{\mathrm{alg}}$ introduced by G. Wilson and a quotient of the group of tame symplectic automorphisms of the path algebra of a quiver introduced by Bielawski and Pidstrygach. The latter is known to act transitively on the phase space \(\mathcal{C}_{n,2}\) of the Gibbons–Hermsen integrable system of rank $2$, and we prove that the subgroup generated by the image of $\Gamma^{\mathrm{alg}}$ together with a particular tame symplectic automorphism has the property that, for every pair of points of the regular and semisimple locus of \(\mathcal{C}_{n,2}\), the subgroup contains an element sending the first point to the second.