  In this article we report a preliminary investigation of the large $N$ limit
of a generalized one-matrix model which represents an $O(n)$ symmetric model on
a random lattice. The model on a regular lattice is known to be critical only
for $-2\le n\le 2$. This is the situation we shall discuss also here, using
steepest descent. We first determine the critical and multicritical points,
recovering in particular results previously obtained by Kostov. We then
calculate the scaling behaviour in the critical region when the cosmological
constant is close to its critical value. Like for the multi-matrix models, all
critical points can be classified in terms of two relatively prime integers
$p,q$. In the parametrization $p=(2m+1)q \pm l$, $m,l$ integers such that
$0<l<q$, the string susceptibility exponent is found to be $\gamma_{\rm
string}=-2l/(p+q-l)$. When $l=1$ we find that all results agree with those of
the corresponding $(p,q)$ string models, otherwise they are different.\par We
finally explain how to derive the large order behaviour of the corresponding
topological expansion in the double scaling limit.
