Geodesic
Finite-type integrable geometric structures
Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 1, pp. 255-269.

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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. 
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V. A. Yumaguzhin. Finite-type integrable geometric structures. Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 1, pp. 255-269. http://geodesic.mathdoc.fr/item/FPM_2004_10_1_a12/

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