A new representation of vacuum Lovelock solutions in $d = 2N+1$ dimensions: Black holes with an integrable singularity and regular black holes

In recent years, black hole (BH) solutions with an integrable singularity have garnered significant attention as alternatives to regular black holes (RBH). In these models, similarly to RBHs, an object would not undergo spaghettification when approaching the radial origin. Instead of the potentially unstable de Sitter core present in RBHs, an integrable singularity emerges where the Ricci scalar diverges while its volume integral remains finite. However, the construction of both RBH solutions and BHs with an integrable singularity typically requires the inclusion of specific forms of matter in the energy-momentum tensor. We demonstrate that, from a geometric perspective in the absence of matter, vacuum solutions in Lovelock gravity in $d=2N+1$ dimensions can be represented as vacuum BHs with an integrable singularity in Einstein-Gauss-Bonnet theory for $d=5$ and in cubic gravity for $d=7$. Meanwhile, the vacuum solution in quartic gravity in $d=9$ is described as a vacuum RBH with a nontrivial hyperboloidal cross-section. For all the aforementioned cases, we have determined the conditions that the parameters in the solutions must satisfy. Remarkably, in all discussed cases, there is no presence of an internal horizon near a potentially unstable de Sitter core.