Local solvability for top degree forms in a class of systems of vector fields

We study the local exactness at the top degree level in the differential complex defined by a smooth, locally integrable structure of rank n in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. If Z denotes a local first integral of the structure it is proved that the vanishing of the local cohomology in degree n is implied by the absence of compact connected components of the "fibers" Z = const. This adds one more result towards the verification of a conjecture due to F. Treves regarding the vanishing of the local cohomology of such complexes of differential operators.