  Let X:R2\Dr->R2 be a differentiable (but not necessarily C1) vector field,
where r>0 and Dr={z\in R2:|z|\le r}. If for some e>0 and for all p\in R2\Dr, no
eigenvalue of D_p X belongs to (-e,0]\cup {z\in\C:\mathcal{R}(z)\ge 0}, then
(a)For all p\in R2\Dr, there is a unique positive semi--trajectory of X
starting at p; (b)\mathcal{I}(X), the index of X at infinity, is a well defined
number of the extended real line [-\infty,\infty); (c) There exists a constant
vector v\in R2 such that if \mathcal{I}(X) is less than zero (resp. greater or
equal to zero), then the point at infinity \infty of the Riemann sphere
R2\cup\set{\infty} is a repellor (resp. an attractor) of the vector field X+v.
