  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 $r\geq r_{\text{ms}}$ (the
minimally stable radius of particle orbits), while the radii of singularity/
throat/ horizon $r<r_{\text{ms}}$. Also, its Page-Thorne efficiency $\epsilon$
is found to increase with decreasing $r_{\text{ms}}$ and also yields
$\epsilon=0.0572$ for Schwarzschild black hole (SBH). But in the singular limit
$r\rightarrow r_{s}$ (radius of singularity), we have $\epsilon\rightarrow 1$
giving rise to $100 \%$ efficiency in agreement with the efficiency of the
naked singularity constructed in [10]. We show that the differential accretion
luminosity $\frac{d\mathcal{L}_{\infty}}{d\ln{r}}$ of Buchdahl naked
singularity (BNS) is always substantially larger than that of SBH, while
Eddington luminosity at infinity $L_{\text{Edd}}^{\infty}$ for BNS could be
arbitrarily large at $r\rightarrow r_{s}$ due to the scalar field $\phi$ that
is defined in $(r_{s}, \infty)$. It is concluded that BNS accretion profiles
can still be higher than those of regular objects in the universe.
