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

We show that there exists a morphism between a group ${\mathrm{\Gamma }}^{\mathrm{a}\mathrm{l}\mathrm{g}}$ 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 ${\mathrm{\Gamma }}^{\mathrm{a}\mathrm{l}\mathrm{g}}$ 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.