Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow $\Phi$ of $X$ is defined. The uniquely determined maximal such $U$ can be obtained as the union of all the initial conditions with the maximal time interval on which the particular solution curve is defined, i.e. $U = \bigcup_{p \in M} \{p\} \times I_p$ where $I_p \subseteq \mathbb{R}$ is the maximal open interval around $t = 0$ where $\Phi(p, t)$ is an integral curve of $X$ with initial condition $p$.
Now it is fairly easy to see that $U$ has certain properties like: if there is a non-zero $\epsilon$ such that $M \times (-\epsilon, \epsilon) \subseteq U$ then $U = M \times \mathbb{R}$, i.e. if the solutions are defined for a common small time uniformly for all initial conditions then the flow is already complete, a feature which is of course very nice to have. If $M$ is compact, this is of course always fulfilled.
My question is now how one can characterize the possible domains of in-complete flows: is the above condition the only thing which has to fail for $U$ in order to find a vector field $X$ such that its flow has precisely $U$ as domain for the flow? Or are there other conditions on $U$ which makes $U$ a domain of a flow beyond the above necessary "non-uniform width"?
