A criterion for completeness of an analytic vector field on a locally compact Abelian group

We give a criterion for completeness of a compactly supported analytic vector field $C$ on a locally compact Abelian group $X$. It is shown that $C$ gives rise to an analytic flow in $X$ if and only if there exists on $X$ a fundamental family of cylindrical differential systems (Frobenius distributions) that are stable with respect to the action of $C$.