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

We investigate the finite dimensional dynamical system derived by Braden and Hone in 1996 from the solitons of An−1 affine Toda field theory. This system of evolution equations for an Hermitian matrix L and a real diagonal matrix q with distinct eigenvalues was interpreted as a special case of the spin Ruijsenaars-Schneider models due to Krichever and Zabrodin. A decade later, Li re-derived the model from a general framework built on coboundary dynamical Poisson groupoids. This led to a Hamiltonian description of the gauge invariant content of the model, where the gauge transformations act as conjugations of L by diagonal unitary matrices. Here, we point out that the same dynamics can be interpreted also as a special case of the spin Sutherland systems obtained by reducing the free geodesic motion on symmetric spaces, studied by Pusztai and the author in 2006; the relevant symmetric space being . This construction provides an alternative Hamiltonian interpretation of the Braden-Hone dynamics. We prove that the two Poisson brackets are compatible and yield a bi-Hamiltonian description of the standard commuting flows of the model.