  We study the complete K\"{a}hler-Einstein metric of a Hartogs domain
$\widetilde {\Omega}$, which is obtained by inflation of an irreducible bounded
symmetric domain $\Omega $, using a power $N^{\mu}$ of the generic norm of
$\Omega$. The generating function of the K\"{a}hler-Einstein metric satisfies a
complex Monge-Amp\`{e}re equation with boundary condition. The domain
$\widetilde {\Omega}$ is in general not homogeneous, but it has a subgroup of
automorphisms, the orbits of which are parameterized by $X\in\lbrack0,1[$. This
allows to reduce the Monge-Amp\`{e}re equation to an ordinary differential
equation with limit condition. This equation can be explicitly solved for a
special value $\mu_{0}$ of $\mu$, called the critical exponent. We work out the
details for the two exceptional symmetric domains. The critical exponent seems
also to be relevant for the properties of other invariant metrics like the
Bergman metric; a conjecture is stated, which is proved for the exceptional
domains.
