Uniqueness of the Jackiw non-Noetherian conformal scalar field

Jackiw was undoubtedly the first to exhibit an example of a scalar field action which is not conformally invariant whereas its equation of motion is. This feature has recently been dubbed as a non-Noetherian conformal scalar field. The paradigmatic example of Jackiw was the generalization to curved spacetime of the two-dimensional Liouville action. Here, we prove that, up to second order, this is the unique example of a non-Noetherian conformal scalar field in two dimensions. We establish this result using an old and somewhat forgotten theorem which is none other than the solution to the inverse problem of the calculus of variations.Comment: We dedicate this note to Prof. Roman Jackiw, the pioneer in the exploration of non-Noetherian conformal scalar fields and a champion of subtle thinking in the wonderland of symmetrie