Linear subspaces of smooth vector fields as a kernel of some linear first order partial differential equation

There are given sufficient conditions for linear subspaces S ⊆ V = C ∞ ( G ⊆ R n ; R n ) of smooth vector fields to be written as a kernel of some linear first order partial differential equation (PDE). In particular, it implies that an overdetermined system of linear first order PDE has a nontrivial solution and, in addition, the control system associated with S ⊆ V is not controllable.