  Recently, an extension of the standard four-dimensional scalar conformal
action, yielding a second-order field equation that remains conformally
invariant, was proposed. In spite of this, the corresponding action is not
invariant under conformal transformations and this motivates us to define the
notion of non-Noetherian conformal scalar field. In this article, we go further
by determining the most general action in four dimensions that gives rise to a
non-Noetherian conformal scalar field satisfying a second-order equation. This
task is achieved by using the solution to the inverse problem of the calculus
of variations. Surprisingly enough, the standard equation is shown to be
extended by a non-Noetherian conformal piece involving a nonminimal coupling
with a very particular combination of squared curvature terms, which is none
other than the one defining the so-called Critical Gravity. We also prove that
the most general second-order Euler-Lagrange equation for a conformal scalar
field involves additional Noetherian conformal nonminimal couplings defined by
an arbitrary function of the Weyl tensor. The recently proposed non-Noetherian
conformal extension is recovered as a particular example of this function.
