A note on the automorphism group of the Bielawski-Pidstrygach quiver

We show that there exists a morphism between a group 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 Cn,2 of the Gibbons-Hermsen integrable system of rank 2, and we prove that the subgroup generated by the image alg together with a particular tame symplectic automorphism has the property that, for every pair of points of the regular and semisimple locus of Cn,2, the subgroup contains an element sending the first point to the second.