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

We study the K\"ahler-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.