$\newcommand{\R}{\mathbf R}$ Let $V:U\to \R^n$ be a (continuous) vector field on an open set $U$ of $\R^n$. Suppose we have a point $\mathbf p\in U$ and an open interval $I\subseteq \R$ such that there is an integral curve $\gamma^{\mathbf p}:I\to U$ starting at $0$ (This means that $I$ contains $0$ and $\gamma^{\mathbf p}(0)=\mathbf p$). Let $t_0\in I$.
Does there exist a neighborhood $N$ of $\mathbf p$ and an open interval $J$ containing $0$ and $t_0$ such that for all points $\mathbf q$ of $N$, there is an integral curve $\gamma^{\mathbf q}:J\to U$ starting at $\mathbf q$?
I am aware of the following theorem of which the above is, if true, a stronger version:
Let $V:U\to \R^n$ be a vector field on an open subset $U$ of $\R^n$ and $\mathbf p$ be any point in $U$. Then there exists $\varepsilon>0$ and a neighborhood $N$ of $\mathbf p$ such that for each point $\mathbf q\in N$, there is an integral curve $\gamma^{\mathbf q}:(-\varepsilon, \varepsilon)\to U$.
Does anybody know if the thing in the block quote is true? If so can somebody provide a proof or point me towards a reference?
Thanks.