  In this, paper, we give a complete system of analytic invariants for the
unfoldings of nonresonant linear differential systems with an irregular
singularity of Poincar\'e rank 1 at the origin over a fixed neighborhood $D_r$.
The unfolding parameter $\epsilon $ is taken in a sector S pointed at the
origin of opening larger than $2 \pi$ in the complex plane, thus covering a
whole neighborhood of the origin. For each parameter value in S, we cover $D_r$
with two sectors and, over each sector, we construct a well chosen basis of
solutions of the unfolded linear differential systems. This basis is used to
find the analytic invariants linked to the monodromy of the chosen basis around
the singular points. The analytic invariants give a complete geometric
interpretation to the well-known Stokes matrices at $\epsilon =0$: this
includes the link (existing at least for the generic cases) between the
divergence of the solutions at $\epsilon =0$ and the presence of logarithmic
terms in the solutions for resonance values of the unfolding parameter.
Finally, we give a realization theorem for a given complete system of analytic
invariants satisfying a necessary and sufficient condition, thus identifying
the set of modules.
