  We consider the hermitian matrix model with an external field entering the
quadratic term $\tr(\Lambda X\Lambda X)$ and Penner--like interaction term
$\alpha N(\log(1+X)-X)$. An explicit solution in the leading order in $N$ is
presented. The critical behaviour is given by the second derivative of the free
energy in $\alpha$ which appears to be a pure logarithm, that is a feature of
$c=1$ theories. Various critical regimes are possible, some of them corresponds
to critical points of the usual Penner model, but there exists an infinite set
of multi-critical points which differ by values of scaling dimensions of proper
conformal operators. Their correlators with the puncture operator are given in
genus zero by Legendre polynomials whose argument is determined by an analog of
the string equation.
