Local exactness in a class of differential complexes

The article studies the local exactness at level in the differential complex defined by commuting, linearly independent real-analytic complex vector fields in independent variables. Locally the system admits a first integral , i.e., a complex function such that and . The germs of the ``level sets' of , the sets , are invariants of the structure. It is proved that the vanishing of the (reduced) singular homology, in dimension , of these level sets is sufficient for local exactness at the level . The condition was already known to be necessary.