Let $\tau>0$, $d\in\mathbb N$ and $T_t$ be the $C^1$-diffeomorphism on $\mathbb R^d$ given by $$T_t(x)=x+\int_0^tv(s,T_s(x))\:{\rm d}s\;\;\;\text{for all }x\in\mathbb R^d\tag1$$ for $t\in[0,\tau]$, where $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ with $v(\;\cdot\;,x)$ is continuous for all $x\in\mathbb R^d$ and $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)\right\|\le c\left\|x-y\right\|_E\;\;\;\text{for all }x,y\in E\tag2$$ for some $c\ge0$.
Are we able to extend $(1)$ for negative $t$ (maybe by reflecting the solution somehow) so that $(1)$ holds for all $t\in[-\tau,\tau]$?