  We introduce a new 1-matrix model with arbitrary potential and the
matrix-valued background field. Its partition function is a $\tau$-function of
KP-hierarchy, subjected to a kind of ${\cal L}_{-1}$-constraint. Moreover,
partition function behaves smoothly in the limit of infinitely large matrices.
If the potential is equal to $X^{K+1}$, this partition function becomes a
$\tau$-function of $K$-reduced KP-hierarchy, obeying a set of ${\cal W}
_K$-algebra constraints identical to those conjectured in \cite{FKN91} for
double-scaling continuum limit of $(K-1)$-matrix model. In the case of $K=2$
the statement reduces to the early established \cite{MMM91b} relation between
Kontsevich model and the ordinary $2d$ quantum gravity . Kontsevich model with
generic potential may be considered as interpolation between all the models of
$2d$ quantum gravity with $c<1$ preserving the property of integrability and
the analogue of string equation.
