  We consider solutions of the semi-classical Einstein-Klein-Gordon system with
a cosmological constant $\Lambda\in\mathbb{R}$, where the spacetime is given by
Einstein's static metric on $\mathbb{R}\times\mathbb{S}^3$ with a round sphere
of radius $a>0$ and the state of the scalar quantum field has a two-point
distribution $\omega_2$ that respects all the symmetries of the metric. We
assume that the mass $m\ge0$ and scalar curvature coupling $\xi\in\mathbb{R}$
of the field satisfy $m^2+\xi R>0$, which entails the existence of a ground
state. We do not require states to be Hadamard or quasi-free, but the
quasi-free solutions are characterised in full detail.
  The set of solutions of the semi-classical Einstein-Klein-Gordon system
depends on the choice of the parameters $(a,\Lambda,m,\xi)$ and on the
renormalisation constants in the renormalised stress tensor of the scalar
field. We show that the set of solutions is either (i) the empty set, or (ii)
the singleton set containing only the ground state, or (iii) a set with
infinitely many elements. We characterise the ranges of the parameters and
renormalisation constants where each of these alternatives occur. We also show
that all quasi-free solutions are given by density matrices in the ground state
representation and we show that in cases (ii) and (iii) there is a unique
quasi-free solution which minimises the von Neumann entropy. When $m=0$ this
unique state is a $\beta$-KMS state. We argue that all these conclusions remain
valid in the reduced order formulation of the semi-classical Einstein equation.
