  Some sixty years ago Buchdahl pioneered a program in search of static
spherically symmetric vacua for pure $R^{2}$ gravity (Nuovo Cimento, Vol 23, No
1, pp 141-157 (1962); [https://link.springer.com/article/10.1007/BF02733549]).
Surpassing several obstacles, his work culminated in a non-linear second-order
ordinary differential equation (ODE) which required being solved. However
Buchdahl deemed the ODE intractable and prematurely abandoned his pursuit for
an analytical solution. We have finally managed to break this six-decades-old
impasse and accomplish his goal. Reformulating the Buchdahl ODE, we obtain a
novel exhaustive class of metrics (which we shall call the Buchdahl-inspired
metrics hereafter) in a compact form. We are able to verify ex post, via direct
inspection, that the metric obtained herein satisfies the $R^{2}$ vacuo field
equation, hence establishing its validity. We show why the Buchdahl-inspired
metrics, which exhibit non-constant scalar curvature, defeat a
Lichnerowicz-type no-go theorem, previously proved for quadratic gravity, by
evading the overly strong restriction on the asymptotic falloff of the metric
at spatial infinity. Our novel solution thereby completes Buchdahl's
six-decades-old program. We also explore the mathematical properties of the
Buchdahl-inspired metrics in the limit of small k and in the region around the
coordinate origin.
