  The gravitational Cheshire effect refers to the possibility of turning off
the gravitational field while still leaving an imprint of the nonminimal
coupling of matter to gravity. This allows nontrivial solutions in flat
spacetime for which no backreaction is possible. The effect was originally
shown to manifest itself for standard nonminimal couplings, such as those
allowing conventional conformally invariant scalar fields. Recently, the most
general scalar field action yielding a conformally invariant second-order
equation was constructed, and entails a more involved nonminimal coupling
explicitly breaking the conformal invariance of the action without spoiling it
in the equation. We have succeeded in fully describing the spherically
symmetric stealth solutions on flat spacetime supporting the Cheshire effect
within this general non-Noetherian conformal theory. The allowed configurations
are divided into two branches: The first one essentially corresponds to an
extension of the solutions already known for the standard Noetherian conformal
theory. The second branch is only possible due to the non-Noetherian conformal
contribution of the action. The complete characterization of this branch is
expressed by a nonlinear first-order partial differential equation. We have
found the general solution of this equation using both seemingly new and
well-established mathematical tools.
