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We prove two rigidity results on holomorphic isometries into homogeneous K\"{a}hler manifolds. The first shows that a K\"{a}hler-Ricci soliton induced by the homogeneous metric of the K\"{a}hler product of a special flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. K\"{a}hler-Einstein. In the second one we prove that: (i) a flat space is not relative to the K\"{a}hler product of a special flag manifold with a homogeneous bounded domain, (ii) a special flag manifold is not relative to the K\"{a}hler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the K\"{a}hler product of a flat space with a special flag manifold. Our theorems strongly extend the results in [4], [5], [12], [13] and [22]., Comment: 14 pages

Let $(M, g)$ be a Kaehler manifold whose associated Kaehler form $\omega$ is integral and let $(L, h)\rightarrow (M, \omega)$ be a quantization hermitian line bundle. In this paper we study those Kaehler manifolds $(M, g)$ admitting a finite TYCZ expansion. We show that if the TYCZ expansion is finite then $T_{mg}$ is indeed a polynomial in $m$ of degree $n$, $n=dim M$, and the log-term of the Szeg\"{o} kernel of the disc bundle $D\subset L^*$ vanishes (where $L^*$ is the dual bundle of $L$). Moreover, we provide a complete classification of the Kaehler manifolds admitting finite TYCZ expansion either when $M$ is a complex curve or when $M$ is a complex surface with a cscK metric which admits a radial Kaehler potential.