Generalized Toda Mechanics Associated with Classical Lie Algebras and Their Reductions

For any classical Lie algebra $\mathfrak{g}$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for $\mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,n\leq3$ are also given. For all $m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Toda chains are natural reductions of that of the $(m,n)$-chain, and for $m=n$, there is also a family of symmetrically reduced Toda systems, the $(m,m)_{\mathrm{Sym}}$-Toda systems, which are also integrable. In the quantum case, all $(m,n)$-Toda systems with $m>1$ or $n>1$ describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all $(m,n)$-Toda systems survive after quantization.