  We study the large N limit of the Itzykson -- Zuber integral and show that
the leading term is given by the exponent of an action functional for the
complex inviscid Burgers (Hopf) equation evaluated on its particular classical
solution; the eigenvalue densities that enter in the IZ integral being the
imaginary parts of the boundary values of this solution. We show how this
result can be applied to ``induced QCD" with an arbitrary potential $U(x)$. We
find that for a nonsingular $U(x)$ in one dimension the eigenvalue density
$\rho(x)$ at the saddle point is the solution of the functional equation
$G_{+}(G_{-}(x))=G_{-}(G_{+}(x))=x$, where $G_{\pm}(x) \equiv
{1\over{2}}U^{\prime}(x)\pm i\pi \rho(x)$. As an illustration we present a
number of new particular solutions of the $c=1$ matrix model on a discrete real
line.
