Abstract
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 slntor. At last, the relation of these two kinds of extensions also is discussed.
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