Higher Toda Mechanics and Spectral Curves

For each of the Lie algebras $mathfrak{gl}_{n}$ and$widetilde{mathfrak{gl}}_{n}$, we construct a family of integrable generalizations ofthe Toda chains characterized by two integers $m_{+}$ and $m_{-}$.The Lax matrices and the equations of motion are given explicitly, and the integralsof motion can be calculated in terms of the trace of powers of the Lax matrixL. For the case of $m_{+}=m_{-}$, we find a symmetric reduction for eachgeneralized Toda chain we found, and the solution to the initial valueproblems of the reduced systems is outlined. We also studied the spectralcurves of the periodic $(m_{+},m_{-})$-Toda chains, which turns out to be verydifferent for different pairs of $m_{+}$ and $m_{-}$. Finally we also obtainthe nonabelian generalizations of the $(m_{+},m_{-})$-Toda chains in an explicit form.