  In violation of the generalized Lichnerowicz theorem advocated by Nelson and
others, quadratic gravity admits vacua with non-constant scalar curvature. In a
recent publication [Phys. Rev. D 106, 104004 (2022)], we revitalized a program
that Buchdahl originated but prematurely abandoned circa 1962 [Nuovo Cimento
23, 141 (1962)], and uncovered a novel exhaustive class of static spherically
symmetric vacua for pure $R^2$ gravity. The Buchdahl-inspired metrics we
obtained therein are exact solutions which exhibit non-constant scalar
curvature. A product of fourth-order gravity, the metrics entail a new
(Buchdahl) parameter $k$ which allows the Ricci scalar to vary on the manifold.
The metrics are able to defeat the generalized Lichnerowicz theorem by evading
an overly strong restriction on the asymptotic falloff in the Ricci scalar as
assumed in the theorem. The Buchdahl parameter $k$ is a new characteristic of
pure $R^2$ gravity, a higher-derivative theory. By venturing that the Buchdahl
parameter should be a universal hallmark of higher-derivative gravity at large,
in this paper we seek to extend the concept to the quadratic action
$R^2+\gamma\,(R-2\Lambda)$. We are able to determine that the quadratic field
equation admits a perturbative vacuo that is valid up to the order $O(k^2)$.
Conforming with our guiding intuition, the Ricci scalar is non-constant,
including the asymptotically flat case, as long as $k\neq0$ and $\gamma\neq0$.
The existence of such an asymptotically flat vacuo with non-constant scalar
curvature defeats the generalized Lichnerowicz theorem in its entirety. Our
finding thus warrants restoring the $R^2$ term in the full quadratic action,
$\gamma\,R+\beta\,R^2-\alpha\,C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}$, when
applying the L\"u-Perkins-Pope-Stelle ansatz. Implications to the
L\"u-Perkins-Pope-Stelle solution are discussed herein.
