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In a recent article [Phys. Rev. D $\textbf{107}$, 104008 (2023)], we reportedthe discovery of a new exact closed analytical solution in pure $R^2$ gravitythat describes asymptotically flat spacetimes. The solution is Ricci scalarflat, viz. $R\equiv0$ everywhere. Whereas any metric with a null Ricci scalarwould $\textit{trivially}$ satisfy the $R^2$ vacuo field equation,$R\left(R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}\,R\right)+g_{\mu\nu}\,\square\,R-\nabla_{\mu}\nabla_{\nu}R=0$,we shall demonstrate that our solution satisfies a ``stronger'' version of the$R^2$ vacuo field equation, viz.$R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}\,R+R^{-1}\left(g_{\mu\nu}\,\square\,R-\nabla_{\mu}\nabla_{\nu}R\right)=0$,despite the term $R^{-1}$ being $\textit{singular}$. Even though the Ricciscalar $R$ identically vanishes, for our solution, the combinations$\,R^{-1}\,\nabla_{\mu}\nabla_{\nu}R\,$ and $\,R^{-1}\,\square\,R\,$ are$\textit{free of singularity}$. This exceptional property sets our solutionapart from the set of null-Ricci-scalar metrics and makes it a genuinely$\textit{non-trivial}$ solution. We further show that, as a member of a largerclass of asymptotically de Sitter metrics, our solution is resilient againstperturbations in the scalar curvature at largest distances, making it relevantfor physical situations where the background deviates from asymptotic flatness.

Non-triviality of asymptotically flat Buchdahl-inspired metrics in pure $R^2$ gravity