  Given an $\widetilde n$-dimensional manifold $\widetilde M$ equipped with a
$\widetilde G$-structure $\widetilde\pi:\widetilde P\rightarrow \widetilde M$,
there is a naturally induced $G$-structure $\pi: P\rightarrow M$ on any
submanifold $M\subset\widetilde M$ that satisfies appropriate regularity
conditions. We study generalized integrability problems for a given
$G$-structure $\pi: P\rightarrow M$, namely the questions of whether it is
locally equivalent to induced $G$-structures on regular submanifolds of
homogeneous $\widetilde G$-structures $\widetilde\pi:\widetilde P\to
\widetilde{H}/\widetilde{K}$. If $\widetilde\pi:\widetilde P\to
\widetilde{H}/\widetilde{K}$ is flat $k$-reductive we introduce a sequence of
generalized curvatures taking values in appropriate cohomology groups and prove
that the vanishing of these curvatures are necessary and sufficient conditions
for the solution of the corresponding generalized integrability problems.
