Each point is contained in precisely one integral curve

Question

As Corollary 2.2.1 in picture below, each point is contained in precisely one integral curve. But it is obvious there are different integral curve which contain the point. I understand 'precisely one' as 'only one', whether my understand is wrong ?

Answer 1

I think some of the confusion may come from the following. Given an integral curve $\gamma: \Bbb R \to M$ for the vector field $X$, then its translates $\gamma_t$, defined by $\gamma_t(s)=\gamma(t+s)$, are also integral curves. The difference is just where you define the starting point of your curve to be.

So this is a whole family of curves. But they have the exact same image, and only differ by (an incredibly simple) reparameterization. When the author says that a point lies in one and only one integral curve, he's identifying the translates above and considering them the same integral curve.