Local exactness in a class of differential complexes

The article studies the local exactness at level $q$ $(1\le q\le n)$ in the differential complex defined by $n$ commuting, linearly independent real-analytic complex vector fields $L_1,\dotsc ,L_n$ in $n+1$ independent variables. Locally the system $\{L_1,\dotsc ,L_n\}$ admits a first integral $Z$, i.e., a $\mathcal {C}^\omega$ complex function $Z$ such that $L_1Z=\cdots =L_nZ=0$ and $dZ\ne 0$. The germs of the ``level sets'' of $Z$, the sets $Z=z_0\in \mathbb {C}$, are invariants of the structure. It is proved that the vanishing of the (reduced) singular homology, in dimension $q-1$, of these level sets is sufficient for local exactness at the level $q$. The condition was already known to be necessary.