  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.
