  A unitary gravitational action up to third order of curvature in which
respects to the holographic $a-$theorem has been constructed in \cite{myers}.
In particular, its third order term is just the Weyl-cubed term in four
dimensions.
  In this paper, we study this theory and find some its exact classical
solutions. We show that the theory admits conformally flat, Lifshitz,
Schr\"{o}dinger and also hyperscaling-violating backgrounds as the solutions of
equations of motion. Our analysis has been done for the pure Weyl-cubed
gravity, Einstein plus Weyl-cubed term and gravity with matter.
