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We investigate the finite dimensional dynamical system derived by Braden andHone in 1996 from the solitons of $A_{n-1}$ affine Toda field theory. Thissystem of evolution equations for an $n\times n$ Hermitian matrix $L$ and areal diagonal matrix $q$ with distinct eigenvalues was interpreted as a specialcase of the spin Ruijsenaars--Schneider models due to Krichever and Zabrodin. Adecade later, L.-C. Li re-derived the model from a general framework built oncoboundary dynamical Poisson groupoids. This led to a Hamiltonian descriptionof the gauge invariant content of the model, where the gauge transformationsact as conjugations of $L$ by diagonal unitary matrices. Here, we point outthat the same dynamics can be interpreted also as a special case of the spinSutherland systems obtained by reducing the free geodesic motion on symmetricspaces, studied by Pusztai and the author in 2006; the relevant symmetric spacebeing $\mathrm{GL}(n,\mathbb{C})/ \mathrm{U}(n)$. This construction provides analternative Hamiltonian interpretation of the Braden--Hone dynamics. We provethat the two Poisson brackets are compatible and yield a bi-Hamiltoniandescription of the standard commuting flows of the model.

Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone