Multiple Hamiltonian structures for Toda systems of type \(A\)-\(B\)-\(C\) (Q1577149)

A Toda lattice is a system of particles on the line with exponential interaction [\textit{M.~Toda}, J. Phys. Soc. Jap. 22, 431-436 (1967)]. The equations for the Toda systems admit Lax representations. The author shows that Toda systems of type \(B_n\) and \(C_n\) are bi-Hamiltonian. He also constructs a recursion operator and master symmetries. The main result is as follows: The degrees of the higher Poisson brackets for \(A\), \(B\) and \(C\)-Toda systems coincide with the exponents of the corresponding Lie groups.