  Let $\alpha$ be an irrational number and $I$ an interval of $\mathbb{R}$. If
$\alpha$ is Diophantine, we show that any one-parameter group of homeomorphisms
of $I$ whose time-$1$ and $\alpha$ maps are $C^\infty$ is in fact the flow of a
$C^\infty$ vector field. If $\alpha$ is Liouville on the other hand, we
construct a one-parameter group of homeomorphisms of $I$ whose time-$1$ and
$\alpha$ maps are $C^\infty$ but which is not the flow of a $C^2$ vector field
(though, if $I$ has boundary, we explain that the hypotheses force it to be the
flow of a $C^1$ vector field). We extend both results to families of irrational
numbers, the critical arithmetic condition in this case being simultaneous
"diophantinity". For one-parameter groups defining a free action of
$(\mathbb{R},+)$ on $I$, these results follow from famous linearization
theorems for circle diffeomorphisms. The novelty of this work concerns non-free
actions.
