  We introduce a family of quivers $Z_{r}$ (labeled by a natural number $r\geq
1$) and study the non-commutative symplectic geometry of the corresponding
doubles $\mathbf{Q}_{r}$. We show that the group of non-commutative
symplectomorphisms of the path algebra $\mathbb{C}\mathbf{Q}_{r}$ contains two
copies of the group $\mathrm{GL}_{r}$ over a ring of polynomials in one
indeterminate, and that a particular subgroup $\mathcal{P}_{r}$ (which contains
both of these copies) acts on the completion $\mathcal{C}_{n,r}$ of the phase
space of the $n$-particles, rank $r$ Gibbons-Hermsen integrable system and
connects each pair of points belonging to a certain dense open subset of
$\mathcal{C}_{n,r}$. This generalizes some known results for the cases $r=1$
and $r=2$.
