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 $L_1(n{)}^M=L_2(n{)}^N+{\sum }_{l\in \mathbb{Z}}{\sum }_{i=1}^m{q}_{i,n}{\mathrm{\Lambda }}^l{r}_{i,n+1}\mathrm{\Delta }$ 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 ${\mathfrak{L}}^k=({\mathfrak{L}}^k{)}_{\ge 1}+{\sum }_{i=1}^m{\mathfrak{q}}_i{\mathrm{\partial }}^{-1}{\mathfrak{r}}_i\mathrm{\partial }$. Next the relation with the Toda reduction ${\mathcal{L}}_1(n{)}^M={\mathcal{L}}_2(n{)}^N+{\sum }_{l\in \mathbb{Z}}{\sum }_{i=1}^m{\tilde{q}}_{i,n}{\mathrm{\Lambda }}^l{\tilde{r}}_{i,n}$ is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.