In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals 
. It is named for Élie Cartan and Erich Kähler.
Meaning
It is not true that merely having 
contained in 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.
Statement
Let 
be a real analytic EDS. Assume that 
is a connected, 
-dimensional, real analytic, regular integral manifold of with 
(i.e., the tangent spaces 
are "extendable" to higher dimensional integral elements).
Moreover, assume there is a real analytic submanifold 
of codimension 
containing 
and such that 
has dimension 
for all 
.
Then there exists a (locally) unique connected, 
-dimensional, real analytic integral manifold 
of that satisfies 
.
Proof and assumptions
The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.