I don’t think your characterization of the vector field is quite correct; I would recommend Cartan ([1922](https://www.emis.de/cgi-bin/JFM-item?48.0538.02), §§[184-185](https://archive.org/stream/leonssurlesinv00cartuoft#page/186)), Godbillon ([1969](http://www.ams.org/mathscinet-getitem?mr=242081), Proposition XI.3.7), or Souriau ([1970](http://www.ams.org/mathscinet-getitem?mr=1461545), Theorem [7.29](https://books.google.com/books?id=_Y3hBwAAQBAJ&pg=PA66)). Or [this answer](https://mathoverflow.net/a/126298) conveniently linked in this page’s “related” list $\to$.