Description
We investigate static, spherically symmetric black holes in regularised four-dimensional Einstein–scalar–Gauss–Bonnet gravity, with the aim of determining whether the theory admits non-asymptotically flat solutions beyond the known special cases. Working in standard spherical coordinates, we use a Frobenius-type expansion of the symmetry-reduced field equations to classify possible local branches of the metric functions and scalar field. While the h=1 sector reproduces the known analytic family, the generic spherical gauge reveals a highly constrained but potentially viable candidate branch. We discuss the structure of this branch, its finite-order consistency checks, and the remaining problem of deriving the full coefficient recursion. Although preliminary, these results indicate a concrete semi-analytical route toward identifying (or ruling out) a more general static spherical black hole family within regularised 4DEsGB gravity.