Moduli space of unfolded differential linear systems with an irregular singularity of Poincaré rank 1

In our recent paper we have identified the moduli space of generic unfoldings of linear differential systems with a nonresonant irregular singularity of Poincaré rank 1 for classification under analytic equivalence. The modulus of the unfolding of a linear differential system is the unfolding of the modulus of the system. It consists in formal invariants and an unfolding of the Stokes matrices. In the realization part, we have identified the realizable moduli. However, the necessary and sufficient condition for realizing unfoldings of Stokes matrices was quite obscure. In this paper we explore this condition and we determine the realizable moduli depending analytically on the parameter in dimensions 2 and 3. In dimension 2, all realizable unfoldings of Stokes matrices can be chosen depending analytically on the parameter. In dimension 3, not all pairs of Stokes matrices have realizable analytic unfoldings.