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F. Podest\`a and A. Spiro introduced a class of $G$-manifolds $M$ with acohomogeneity one action of a compact semisimple Lie group $G$ which admit aninvariant Kaehler structure $(g,J)$ (``standard $G$-manifolds") and studiedinvariant Kaehler and Kaehler-Einstein metrics on $M$. In the first part ofthis paper, we gave a combinatoric description of the standard non compact$G$-manifolds as the total space $M_{\varphi}$ of the homogeneous vector bundle$M = G\times_H V \to S_0 =G/H$ over a flag manifold $S_0$ and we gave necessaryand sufficient conditions for the existence of an invariant Kaehler-Einsteinmetric $g$ on such manifolds $M$ in terms of the existence of an interval inthe $T$-Weyl chamber of the flag manifold $F = G \times _H PV$ which satisfiessome linear condition. In this paper, we consider standard cohomogeneity onemanifolds of a classical simply connected Lie group $G = SU_n, Sp_n. Spin_n$and reformulate these necessary and sufficient conditions in terms of easilychecked arithmetic properties of the Koszul numbers associated with the flagmanifold $S_0 = G/H$. If this conditions is fulfilled, the explicitconstruction of the Kaehler-Einstein metric reduces to the calculation of theinverse function to a given function of one variable.

Cohomogeneity one Kaehler and Kaehler-Einstein manifolds with one singular orbit, II