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In this, paper, we give a complete system of analytic invariants for theunfoldings of nonresonant linear differential systems with an irregularsingularity 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 theorigin of opening larger than $2 \pi$ in the complex plane, thus covering awhole 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 ofsolutions of the unfolded linear differential systems. This basis is used tofind the analytic invariants linked to the monodromy of the chosen basis aroundthe singular points. The analytic invariants give a complete geometricinterpretation to the well-known Stokes matrices at $\epsilon =0$: thisincludes the link (existing at least for the generic cases) between thedivergence of the solutions at $\epsilon =0$ and the presence of logarithmicterms in the solutions for resonance values of the unfolding parameter.Finally, we give a realization theorem for a given complete system of analyticinvariants satisfying a necessary and sufficient condition, thus identifyingthe set of modules.

Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincar\'e rank 1