  In this paper we expand on the work of the first author on ambient
obstruction solitons, which are self-similar solutions to the ambient
obstruction flow. Our main result is to show that any closed ambient
obstruction soliton is ambient obstruction flat and has constant scalar
curvature. We show, in fact, that the first part of this result is true for a
more general extended soliton equation where we allow an arbitrary conformal
factor to be added to the equation. We discuss how this implies that, on a
compact manifold, the ambient obstruction flow has no fixed points up to
conformal diffeomorphism other than ambient obstruction flat metrics. These
results are the consequence of a general integral inequality that can be
applied to the solitons to any geometric flow. Additionally, we use these
results to obtain a generalization of the Bourguignon-Ezin identity on a closed
Riemannian manifold and study the converse problem on a closed extended
$q$-soliton. We also study this extended equation further in the case of
homogeneous and product metrics.
