In his monograph “Systèmes Orthogonaux” (Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910), Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of PDEs, where for each unknown function a certain subset of partial derivatives is prescribed and the values of the unknown functions are prescribed along the corresponding transversal coordinate affine subspaces.  The more general of the theorems, Théorème III, was proved by Darboux only for the cases of 2 and 3 independent variables. In a recent paper published in the Journal of Geometric Analysis, Michael Benfield, Kris Jenssen and myself formulate and prove a generalization of Théorème III. Instead of partial derivatives, we allow to prescribe derivatives of the unknown functions along vector fields. The values of the unknown functions are prescribed along  arbitrary transversal submanifolds. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the set of vector fields and on the initial manifolds. The SCC is automatically
met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a  C^1-solution via Picard iteration for any number of independent variables. If the SCC is not satisfied, we show by an example that the uniqueness may fail.