The M-truncated KP hierarchy and matrix models

Inspired by the Kontsevich model we study the one-matrix model in an external field having the most general potential, called the generalized Kontsevich model. The partition function is shown to satisfy the Virasoro constraints, which are of the different kind to those discussed for the original Kontsevich model. In a special case it turns out to be identical to the partition function of the rectangular matrix model in the double scaling limit. The Virasoro constraints take a simple form and encode all the informations on critically of the latter model.