  Very recently Harada proposed a gravitational theory which is of third order
in the derivatives of the metric tensor with the property that any solution of
Einstein's field equations (EFEs) possibly with a cosmological constant is
necessarily a solution of the new theory. He then applied his theory to derive
a second-order ODE for the evolution of the scale factor of the FLRW metric.
Remarkably he showed that, even in a matter-dominated universe with zero
cosmological constant, there is a late-time transition from decelerating to
accelerating expansion. Harada also derived a generalisation of the
Schwarzschild solution. However, as his starting point he assumed an
unnecessarily restricted form for a static spherically symmetric metric. In
this note the most general spherically symmetric static vacuum solution of the
theory is derived.
  Mantica and Molinari have shown that Harada's theory may be recast into the
form of the EFEs with an additional source term in the form of a second-order
conformal Killing tensor(CKT). Accordingly they have dubbed the theory
conformal Killing gravity. Then, using a result in a previous paper of theirs
on CKTs in generalised Robertson-Walker spacetimes, they rederived Harada's
generalised evolution equation for the scale factor of the FLRW metric.
  However, Mantica and Molinari appear to have overlooked the fact that all
solutions of the new theory (except those satisfying the EFEs) admit a
non-trivial second-order Killing tensor. Such Killing tensors are invaluable
when considering the geodesics of a metric as they lead to a second quadratic
invariant of the motion in addition to that derived from the metric.
