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We study the complete K\"{a}hler-Einstein metric of a Hartogs domain$\widetilde {\Omega}$, which is obtained by inflation of an irreducible boundedsymmetric domain $\Omega $, using a power $N^{\mu}$ of the generic norm of$\Omega$. The generating function of the K\"{a}hler-Einstein metric satisfies acomplex Monge-Amp\`{e}re equation with boundary condition. The domain$\widetilde {\Omega}$ is in general not homogeneous, but it has a subgroup ofautomorphisms, the orbits of which are parameterized by $X\in\lbrack0,1[$. Thisallows to reduce the Monge-Amp\`{e}re equation to an ordinary differentialequation with limit condition. This equation can be explicitly solved for aspecial value $\mu_{0}$ of $\mu$, called the critical exponent. We work out thedetails for the two exceptional symmetric domains. The critical exponent seemsalso to be relevant for the properties of other invariant metrics like theBergman metric; a conjecture is stated, which is proved for the exceptionaldomains.Comment: 37 page

The Kähler-Einstein metric for some Hartogs domains over symmetric domains