  We touch upon a long-standing question of the "true" one-dimensional hydrogen
atom solution. From a symmetry point of view, Kepler problem in $d\ge2$
dimension is characterized by geometrical rotational symmetry, $SO(d)$, as well
as dynamical, "accidental" $SO(d+1)$ symmetry. Because of topology, these two
symmetries are mutually exclusive in one dimension, regardless of the
regularization employed, drawing one to a conclusion that the question of
"true" hydrogen atom in one dimension doesn't have an answer because a single
dimension can not support both of the symmetries of Kepler problem. We argue
our findings using a novel method to recover and classify solutions appearing
in the literature according to the symmetry they respect. In particular,
curious features of some of the solutions - double degeneracy and particle
confinement - are directly attributed to the dynamical symmetry behind them.
