  This paper investigates Buchdahl transformations within the framework of
Einstein and Einstein-Scalar theories. Specifically, we establish that the
recently proposed Schwarzschild-Levi-Civita spacetime can be obtained by means
of a Buchdahl transformation of the Schwarschild metric along the spacelike
Killing vector. The study extends Buchdahl's original theorem by combining it
with the Kerr-Schild representation. In doing so, we construct new
vacuum-rotating black holes in higher dimensions which can be viewed as the
Levi-Civita extensions of the Myers-Perry geometries. Furthermore, it
demonstrates that the double copy scheme within these new generated geometries
still holds, providing an example of an algebraically general double copy
framework. In the context of the Einstein-Scalar system, the paper extends the
corresponding Buchdahl theorem to scenarios where a static vacuum seed
configuration, transformed with respect to a spacelike Killing vector,
generates a hairy black hole spacetime. We analyze the geometrical features of
these spacetimes and investigate how a change of frame, via conformal
transformations, leads to a new family of black hole spacetimes within the
Einstein-Conformal-Scalar system.
