We first advance a mathematical novelty that the three geometrically and topologically distinct objects mentioned in the title can be exactly obtained from the Jordan frame vacuum Brans I solution by a combination of coordinate transformations, trigonometric identities and complex Wick rotation. Next, we study their respective accretion properties using the Page–Thorne model which studies accretion properties exclusively for r ≥ r ms (the minimally stable radius of particle orbits), while the radii of singularity/throat/horizon r < r ms . Also, its Page–Thorne efficiency ϵ is found to increase with decreasing r ms and also yields ϵ = 0.0572 for Schwarzschild black hole (SBH). But in the singular limit r → r s (radius of singularity), we have ϵ → 1 giving rise to 100 % efficiency in agreement with the efficiency of the naked singularity constructed in 10. We show that the differential accretion luminosity d L ∞ d ln r of Buchdahl naked singularity (BNS) is always substantially larger than that of SBH, while Eddington luminosity at infinity L Edd ∞ for BNS could be arbitrarily large at r → r s due to the scalar field ϕ that is defined in ( r s , ∞ ) . It is concluded that BNS accretion profiles can still be higher than those of regular objects in the universe.