The Virasoro symmetries of the bigraded modified Toda hierarchy

Modified Toda hierarchy is a two-component generalization of the 1st modified KP hierarchy, which has been widely applied to analyze constraints of the Toda hierarchy, including the B–Toda and C–Toda hierarchies. In this paper, we construct additional symmetries for the modified Toda hierarchy and derive the corresponding Adler-Shiota-van Moerbeke formula. In addition, we also show that $(N,M)$–bigraded modified Toda hierarchy(BMTH), which is a special reduction for modified Toda, possess additional symmetries that form a subalgebra of the Virasoro algebra.