On the range of unsolvable systems induced by complex vector fields

We provide criteria for local unsolvability of first-order differential systems induced by complex vector fields employing techniques from the theory of locally integrable structures. Following H & ouml;rmander's approach to study locally unsolvable equations, we obtain analogous results in the differential complex associated to a locally integrable structure provided that it is not locally exact in three different scenarios: top-degree, Levi-nondegenerate structures and co-rank 1 structures.