One reduction of the modified Toda hierarchy

The modified Toda (mToda) hierarchy is a two-component generalization of the 1-st modified KP (mKP) hierarchy, which connects the Toda hierarchy via Miura links and has two tau functions. Based on the fact that the mToda and 1-st mKP hierarchies share the same fermionic form, we firstly construct the reduction of the mToda hierarchy $L_1(n)^M=L_2(n)^N+\sum_{l\in\mathbb{Z}}\sum_{i=1}^{m}q_{i,n}\Lambda^lr_{i,n+1}\Delta$ and $(L_1(n)^M+L_2(n)^N)(1)=0$, called the generalized bigraded modified Toda hierarchy, which can be viewed as a new two-component generalization of the constrained mKP hierarchy $\mathfrak{L}^k=(\mathfrak{L}^k)_{\geq 1}+\sum_{i=1}^m \mathfrak{q}_i\partial^{-1}\mathfrak{r}_i\partial$. Next the relation with the Toda reduction $\mathcal{L}_1(n)^M=\mathcal{L}_2(n)^{N}+\sum_{l\in \mathbb{Z}}\sum_{i=1}^{m}\tilde{q}_{i,n}\Lambda^l\tilde{r}_{i,n}$ is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.