Higher Toda Mechanics and Spectral Curves

For each one of the Lie algebras $\mathfrak{gl}_{n}$ and $\widetilde {\mathfrak{gl}}_{n}$, we constructed a family of integrable generalizations of the Toda chains characterized by two integers $m_{+}$ and $m_{-}$. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix $L$. For the case of $m_{+}=m_{-}$, we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic $(m_{+},m_{-})$-Toda chains, which turns out to be very different for different pairs of $m_{+}$ and $m_{-}$. Finally we also obtained the nonabelian generalizations of the $(m_{+},m_{-})$-Toda chains in explicit form.