Symmetry and quaternionic integrable systems

Given a hyperkahler manifold MM, the hyperkahler structure defines a triple of symplectic structures on MM; with these, a triple of Hamiltonians defines a so-called hyperHamiltonian dynamical system on MM. These systems are integrable when can be mapped to a system of quaternionic oscillators. We discuss the symmetry of integrable hyperHamiltonian systems, i.e. quaternionic oscillators, and conversely how these symmetries characterize, at least in the Euclidean case, integrable hyperHamiltonian systems.