  We introduce two classes of "egg type" domains, built on general bounded
symmetric domains, for which we compute the Bergmann kernel in explicit form.
We use the characterization of bounded symmetric domains through Jordan triple
systems. The egg type domains are defined using the generic norm. A
generalization of the Hua integral is computed; the result shows the existence
of a special polynomial with integer or half-integer coefficients, attached to
each irreducible bounded symmetric domain.
