  We prove two rigidity theorems on holomorphic isometries into homogeneous
bounded domains. The first shows that a K\"ahler-Ricci soliton induced by the
homogeneous metric of a homogeneous bounded domain is trivial, i.e.
K\"ahler-Einstein. In the second one we prove that a homogeneous bounded domain
and the flat (definite or indefinite) complex Euclidean space are not
relatives, i.e. they do not share a common K\"ahler submanifold (of positive
dimension). Our theorems extend the results proved in [A. Loi, R. Mossa, Proc.
Amer. Math. Soc. 149 (2021), no. 11, 4931-4941] and [X. Cheng, Y. Hao, Ann.
Global Anal. Geom. 60 (2021), no. 1, 167-180] respectively.

Differential Geometry;Complex Variables