  In this paper we analyze one-matrix models by means of the associated
discrete linear systems. We see that the consistency conditions of the discrete
linear system lead to the Virasoro constraints. The linear system is endowed
with gauge invariances. We show that invariance under time-independent gauge
transformations entails the integrability of the model, while the double
scaling limit is connected with a time-dependent gauge transformation. We
derive the continuum version of the discrete linear system, we prove that the
partition function is actually the $\tau$-function of the KdV hierarchy and
that the linear system completely determines the Virasoro constraints.
