Geodesic
Prolongations of Vector Fields and the Invariants-by-Derivation Property
Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 2, pp. 289-300

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For any given vector field $X$ defined on some open set $M\subset \mathbb R^2$, we characterize the prolongations $X^*_n$ of $X$ to the nth jet space $M^{(n)}$, $n\geq 1$, such that a complete system of invariants for $X^*_n$ can be obtained by derivation of lower-order invariants. This leads to characterizations of $C^{\infty }$-symmetries and to new procedures for reducing the order of an ordinary differential equation.
Keywords: $C^{\infty }$-symmetry, differential invariants, reductions of ordinary differential equations. 
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     author = {C. Muriel and J. L. Romero},
     title = {Prolongations of {Vector} {Fields} and the {Invariants-by-Derivation} {Property}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2002_133_2_a13/}
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C. Muriel; J. L. Romero. Prolongations of Vector Fields and the Invariants-by-Derivation Property. Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 2, pp. 289-300. http://geodesic.mathdoc.fr/item/TMF_2002_133_2_a13/