Geodesic
Homogeneous systems of higher-order ordinary differential equations
Communications in Mathematics, Tome 18 (2010) no. 1, pp. 37-50.

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The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation which is strongly homogeneous in this sense states that the Schwarzian derivative of the dependent variable vanishes. This equation is of importance in the theory of the association between third-order equations and pseudo-Riemannian manifolds due to Newman and his co-workers.
Classification : 34A26, 34C14, 53A55, 53B15, 83C80 
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Crampin, Mike. Homogeneous systems of higher-order ordinary differential equations. Communications in Mathematics, Tome 18 (2010) no. 1, pp. 37-50. http://geodesic.mathdoc.fr/item/COMIM_2010__18_1_a3/