On holomorphic isometries into blow-ups of ${\mathbb{C}}^n$

We study the Kähler-Einstein manifolds which admits a holomorphic isometry into either the generalized Burns-Simanca manifold $({\tilde{\mathbb{C}}}^n,g_S)$ or the Eguchi-Hanson manifold $({\tilde{\mathbb{C}}}^2,g_{EH})$. Moreover, we prove that $({\tilde{\mathbb{C}}}^n,g_S)$ and $({\tilde{\mathbb{C}}}^2,g_{EH})$ are not relatives to any homogeneous bounded domain.