The following definitions and example are taken from John Lee's Smooth Manifolds, 2nd edition.
Given a vector field $V$ on a smooth manifold $M$, we define an integral curve of $V$ to be a differentiable curve $\gamma: J \rightarrow M$ ($J$ being some subinterval of $\Bbb R$) such that for any $t \in J$, $$\gamma'(t) = V_{\gamma(t)}$$ where $\gamma'(t_0) = d\gamma\left(\frac{d}{dt}\Big|_{t_0}\right)$ and $\frac{d}{dt}\Big|_{t_0}$ is the standard basis vector of $T_{t_0}J$. When $0 \in J$, we say that $\gamma(0)$ is the starting point of $\gamma$.
Consider the following example. Let $(x,y)$ be standard coordinates on $\Bbb R^2$, and our vector field $V = \frac{\partial}{\partial x}$. Then the integral curves are subintervals of each straight line parallel to the $x$-axis. (For instance, $J = (-1,1)$, $\gamma(t) = (t,0)$.) However, Lee claims that the integral curves are the entire straight lines - and thus that there is a unique integral curve for each starting point of the plane.
This doesn't mesh with the definition (or the example) above. Have I misunderstood the definition? If so, how?
He's using the (very common) convention that "$J \subset \mathbb{R}$ is an open interval" includes the possibility that $J = \mathbb{R}$ (i.e., $\subset$ means what some other authors might denote by $\subseteq$).
In the example, he should perhaps have said that there is a unique maximal integral curve through each point.