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 L₁(n)M=L₂(n)N+∑ₗ∑ᵢ₌₁ᵐqi,nΛˡri,n+1Δ and (L₁(n)M+L₂(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 Lᵏ=(Lᵏ)≥ 1+∑ᵢ₌₁ᵐ qᵢ∂⁻¹rᵢ∂. Next the relation with the Toda reduction L₁(n)M=L₂(n)N+∑l∈ Z∑ᵢ₌₁ᵐq̃i,nΛˡr̃i,n is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.