On generalized Volterra systems

We construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The algorithm uses the root system of a complex simple Lie algebra. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka–Volterra system. We present in detail all such systems in the cases of A3A3, A4A4 and we also give some examples from higher dimensions. We classify all possible Lotka–Volterra systems that arise via this algorithm in the AnAn case.