  We discuss the existence and non-existence of constant scalar curvature, as
well as extremal, Sasaki metrics. We prove that the natural Sasaki-Boothby-Wang
manifold over the admissible projective bundles over local products of
non-negative CSC K\"ahler metrics, as described in
https://link-springer-com.libproxy.unm.edu/article/10.1007/s00222-008-0126-x,
always has a constant scalar curvature (CSC) Sasaki metric in its Sasaki-Reeb
cone. Moreover, we give examples that show that the extremal Sasaki--Reeb cone,
defined as the set of Sasaki--Reeb vector fields admitting a compatible
extremal Sasaki metric, is not necessarily connected in the Sasaki--Reeb cone,
and it can be empty even in the non-Gorenstein case. We also show by example
that a non-empty extremal Sasaki--Reeb cone need not contain a (CSC) Sasaki
metric which answers a question posed in
https://mathscinet-ams-org.libproxy.unm.edu/mathscinet-getitem?mr=4420789. The
paper also contains an appendix where we explore the existence of K\"ahler
metrics of constant weighted scalar curvature, as defined in
https://londmathsoc-onlinelibrary-wiley-com.libproxy.unm.edu/doi/full/10.1112/plms.12255,
on admissible manifolds over local products of non-negative CSC K\"ahler
metrics.
