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We show how Conformal Gravity (CG) has to satisfy a fine-tuning condition todescribe the rotation curves of disk galaxies without the aid of dark matter.Interpreting CG as a gauge natural theory yields conservation laws and theirassociated superpotentials without ambiguities. We consider the lightdeflection of a point-like lens and impose that the two Schwarzschild-likemetrics with and without the lens are identical at infinite distances from thelens. The energy conservation law implies that the parameter $\gamma$ in thelinear term of the metric has to vanish, otherwise the two metrics arephysically inaccessible from each other. This linear term is responsible tomimic the role of dark matter in disk galaxies and gravitational lensingsystems. Our analysis shows that removing the need of dark matter with CG thusrelies on a fine-tuning condition on $\gamma$. We also illustrate why theresults of previous investigations of gravitational lensing in CG largelydisagree. These discrepancies derive from the erroneous use of the deflectionangle definition adopted in General Relativity, where the vacuum solution isasymptotically flat, unlike CG. In addition, the lens mass is identified withvarious combinations of the metric parameters. However, these identificationsare arbitrary, because the mass is not a conformally invariant quantity, unlikethe conserved charge associated to the energy conservation law. Based on thisconservation law and by removing the fine-tuning condition on $\gamma$, i.e. bysetting $\gamma=0$, the energy difference between the metric with thepoint-like lens and the metric without it defines a conformally invariantquantity that can in principle be used for (1) a proper derivation of lightdeflection in CG, and (2) the identification of the lens mass with a functionof the parameters $\beta$ and $k$ of the Schwarzschild-like metric.

Conformal gravity: light deflection revisited and the galactic rotation curve failure