Isomonodromic deformations along a stratum of the coalescence locus

We consider deformations of a differential system with Poincaré rank 1 at infinity and Fuch-sian singularity at zero along a stratum of a coalescence locus. We give necessary and sufficient conditions for the deformation to be strongly isomonodromic, both as an explicit Pfaffian system (integrable de- formation) and as a non linear system of PDEs on the residue matrix A at the Fuchsian singularity. This construction is complementary to that of [13]. For the specific system here considered, the results generalize those of [26], by giving up the generic conditions, and those of [3], by giving up the Lidskii generic assumption. The importance of the case here considered originates form its applications in the study of strata of Dubrovin-Frobenius manifolds and F -manifolds.