A Cauchy-Kowalevsky theorem for overdetermined systems of nonlinear   partial differential equations and geometric applications
  We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space whose Levi number is maximal. 
M. S. Baouendi, P. Ebenfelt, D. Zaitsev