  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.
