  We examine various examples of horosymmetric manifolds which exhibit
interesting properties with respect to canonical metrics. In particular, we
determine when the blow-up of a quadric along a linear subquadric admits
K\"ahler-Einstein metrics, providing infinitely many examples of manifolds with
no K\"ahler-Ricci solitons that are not K-semistable. Using a different
construction, we provide an infinite family of Fano manifolds with no
K\"ahler-Einstein metrics but which admit coupled K\"ahler-Einstein metrics.
Finally, we elaborate on the relationship between K\"ahler-Ricci solitons and
the more general concept of multiplier Hermitian structures and illustrate this
with examples related to the two previous families.
