Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r. I would like to understand:

1) if there exists an analytic flow $\phi_t(x)$ with complex time $t$ such that $\partial_t \phi_t(x)=v(\phi_t(x))$

2) if such flow is analytic in both $t$ and $x$

3) if the domain of the variable $t$ where $\phi_t(x)$ is analytic is bounded by $r (\sup_{B_r} |v|)^{-1}$ (or something like that).

Are there references about this kind of problems?

Thank you for your attention.