A generalization of an integrability theorem of Darboux

In his monograph Lec{c}ons sur les syst`emes orthogonaux et les coordonnees curvilignes. Principes de geometrie analytique, 1910, Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of the type [partial_{x_i} u_alpha(x)=f^alpha_i(x,u(x)),quad iin I_alphasubseteq{1,dots,n}.] For a given point $bar xin mathbb{R}^n$ it is assumed that the values of the unknown $u_alpha$ are given locally near $bar x$ along ${x,|, x_i=bar x_i , text{for each}, iin I_alpha}$. The more general of the theorems, Theor`eme III, was proved by Darboux only for the cases $n=2$ and $3$. In this work we formulate and prove a generalization of Darbouxs Theor`eme III which applies to systems of the form [{mathbf r}_i(u_alpha)big|_x = f_i^alpha (x, u(x)), quad iin I_alphasubseteq{1,dots,n}] where $mathcal R={{mathbf r}_i}_{i=1}^n$ is a fixed local frame of vector fields near $bar x$. The data for $u_alpha$ are prescribed along a manifold $Xi_alpha$ containing $bar x$ and transverse to the vector fields ${{mathbf r}_i,|, iin I_alpha}$. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame $mathcal R$ and on the manifolds $Xi_alpha$; it 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 $n$.