[For the entire collection see Zbl 0699.00032.] In this interesting paper the authors show that all natural operators transforming every projectable vector field on a fibered manifold Y into a vector field on its r-th prolongation are the constant multiples of the flow operator. Then they deduce an analogous result for the natural operators transforming every vector field on a manifold M into a vector field on any bundle of contact elements over M.
@article{701465,
title = {Prolongation of vector fields to jet bundles},
booktitle = {Proceedings of the Winter School "Geometry and Physics"},
series = {GDML\_Books},
publisher = {Circolo Matematico di Palermo},
address = {Palermo},
year = {1990},
pages = {[103]-111},
mrnumber = {MR1061792},
zbl = {0712.58007},
url = {http://dml.mathdoc.fr/item/701465}
}
Kolář, Ivan; Slovák, Jan. Prolongation of vector fields to jet bundles, dans Proceedings of the Winter School "Geometry and Physics", GDML_Books, (1990), pp. [103]-111. http://gdmltest.u-ga.fr/item/701465/