It is not true that merely having dI contained in I is sufficient for integrability. There is a problem caused by singular solutions. The theorem computes certain constants that must satisfy an inequality in order that there be a solution.
Proof and assumptions
The Cauchy-Kovalevskaya theorem is required, so the analyticity is necessary.
- Jean Dieudonné, Eléments d'analyse, vol. 4, (1977) Chapt. XVIII.13
- R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior Differential Systems, Springer Verlag, New York, 1991.
- Alekseevskii, D.V. (2001), "Pfaffian problem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- E. Cartan, "On the integration of systems of total differential equations," transl. by D. H. Delphenich
- E. Kähler, "Introduction to the theory of systems of differential equations," transl. by D. H. Delphenich