Reading this article, and the definition of one-parameter group, I wonder what naturally occuring (in physics or elsewhere) one-parameter groups $\{F_t:C^\infty(\mathbf{R}^{2n})\to C^\infty(\mathbf{R}^{2n})\}_t$ there are which are not generated by some $\{\bar F_t:\mathbf{R}^{2n}\to\mathbf{R}^{2n}\}_t$, i.e. of the form $\{g\mapsto g\circ\bar F_t\}_t$.
Here is the relevant section:
