  We complement a recent work on the stability of fixed points of the
CMC-Einstein-$\Lambda$ flow. In particular, we modify the utilized gauge for
the Einstein equations and remove a restriction on the fixed points whose
stability we are able to prove by this method, and thereby generalize the
stability result. In addition, we consider the notion of the reduced
Hamiltonian, originally introduced by Fischer and Moncrief for the standard
CMC-Einstein flow. For the analog version of the flow in the presence of a
positive cosmological constant we identify the stationary points and relate
them to the long-time behavior of the flow on manifolds of different Yamabe
types. This entails conjectures on the asymptotic behaviour and potential
attractors.
