Detecting the real line among one-parametric topological groups

We prove that a topological group is isomorphic to the real line if and only if it is a one-parameteric, metrizable, and not monothetic. This result is used in the authors' other paper to prove that one-parametric groups in strictly convex metric group all are topologically isomorphic to the real line. The example of the Bohr topology on the real line demonstrates that metrizability is an essential assumption in our first claim. This motivates further study to characterize the Bohr group topology as well as detect a monothetic one-parametric topological groups in a non-metrizable setting. Both issues are addressed and resolved in the present paper.