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.



Analytic ODE with complex time
You'll find relevant information in the book
Ordinary differential equations in the complex domain
By Einar Hille


<http://books.google.com/books?id=I1OR4t8UZCgC&printsec=frontcover&dq=Ordinary+Differential+Equations+in+the+Complex+Domain&ei=t8wCTPaXFZLKygT_9e24DA&cd=1#v=onepage&q&f=false>


Fixed point (iteration) results are used to prove local existence, and also to give explicit 
lower bounds on the domain of existence, like you want.


