Valery A.Yumaguzhin,

On the obstruction to integrability of almost-complex structures. 

The natural bundle $\pi:E\to M$ of almost-complex structures is considered. 
The action of the pseudogroup of all diffeomorphisms of $M$ on the total space $E$ 
is investigated. A nontrivial 1-st order differential   invariant of this action 
is constructed. It is proved that the Nijenhuise tensor of an almost-complex 
structures is equal to zero iff the constructed invariant for this structure is zero.