Building on a recent work of \v C. Crnkovi��, M. Douglas and G. Moore, a study of multi-critical multi-cut one-matrix models and their associated $sl(2,C)$ integrable hierarchies, is further pursued. The double scaling limits of hermitian matrix models with different scaling ans��tze, lead, to the KdV hierarchy, to the modified KdV hierarchy and part of the non-linear Schr��dinger hierarchy. Instead, the anti-hermitian matrix model, in the two-arc sector, results in the Zakharov-Shabat hierarchy, which contains both KdV and mKdV as reductions. For all the hierarchies, it is found that the Virasoro constraints act on the associated tau-functions. Whereas it is known that the ZS and KdV models lead to the Virasoro constraints of an $sl(2,C)$ vacuum, we find that the mKdV model leads to the Virasoro constraints of a highest weight state with arbitrary conformal dimension.

31 pages