I'm interested in the conditions that a vector field has to satisfy in order for its flow to define a proper action.
Technically, let $\xi$ be a smooth vector field. For each $m\in M$, there is a unique solution $\gamma_m:\mathbb R\mapsto M:\lambda\mapsto\gamma_m(\lambda$) of the differential equation $\frac{d\gamma_m(\lambda)}{d\lambda}=\xi(\gamma_m(\lambda)), \lambda\in\mathbb R$ with initial condition $\gamma_m(0)=m$. The mapping $\varphi_\xi:\mathbb R\times M\mapsto M:(\lambda,m)\mapsto\gamma_m(\lambda)$ is the flow of $\xi$ and defines a left-action of the additive Lie group $G\equiv(\mathbb R,+)$ on $M$.
The action of $\varphi_\xi$ would be said proper if the map $G\times M\mapsto M\times M:(m,\lambda) \mapsto(m,\varphi_\xi(m,\lambda))$ is a proper map.
If anyone has suggestions or references, it would be greatly appreciated.