another topic that is often omitted, but that arnol'd clarifies, is that of the domain of the solution. i.e. if the manifold where then vector field is defiend is compact, then the domain of the solution is all of R, so one gets a 1 parameter flow on the whole manifold, i.e. any point can be flowed along a solution curve for time t, whatever t is. this adds much to the geometry of the subject.