Finite-type integrable geometric structures

In this paper, we consider finite-type geometric structures of arbitrary order and solve the integrability problem for these structures. This problem is equivalent to the integrability problem for the corresponding $G$-structures. The latter problem is solved by constructing the structure functions for $G$-structures of order ${\geq}\,1$. These functions coincide with the well-known ones for the first-order $G$-structures, although their constructions are different. We prove that a finite-type $G$-structure is integrable if and only if the structure functions of the corresponding number of its first prolongations are equal to zero. Applications of this result to second- and third-order ordinary differential equations are noted.