Bigraded modified Toda hierarchy and its extensions

Modified Toda hierarchy is just the two-component first modified KP hierarchy, which is related to 2D Toda hierarchy through Miura transformation and also has been widely used in discussing the B-Toda and C-Toda hierarchies. In this paper, we firstly construct $(N,M)$-bigraded modified Toda hierarchy (BMTH) as a reduction of modified Toda hierarchy, and give corresponding Lie algebra interpretation. After that, we propose two kinds of extensions of $(N,M)$-BMTH. One is extended by using logarithmic flows, while the other is $(2+1)$D extension, which is corresponding to the toroidal Lie algebra ${\mathrm{sl}}_n^{\mathrm{tor}}$. At last, the relation of these two kinds of extensions also is discussed.