The Einstein–Weyl theory is contained as a special case by setting ${\beta=0}$.

*Vacuum field equations* corresponding to the action (\[actionQG\]) are $$\begin{aligned}
&\gamma \left(R_{ab} - {\pul} R\, g_{ab}+\Lambda\,g_{ab}\right)-4 \alpha\,B_{ab} \nonumber \\
&\quad +2\beta\left(R_{ab}-\tfrac{1}{4}R\, g_{ab}+ g_{ab}\, \Box - \nabla_b \nabla_a\right) R = 0 \,, \label{GenQGFieldEq}\end{aligned}$$ where $B_{ab}$ is the *Bach tensor* defined as B\_[ab]{} ( \^c \^d + R\^[cd]{} )C\_[acbd]{} . \[defBach\] It is traceless, symmetric, and conserved: $$g^{ab}B_{ab}=0 \,, \qquad B_{ab}=B_{ba} \,, \qquad \nabla^b B_{ab}=0 \,,
\label{Bachproperties}$$ and also conformally well-behaved (see expression (\[OmBach\]) below).

Now, *assuming* ${R=\hbox{const.}}$, the last two terms in (\[GenQGFieldEq\]) containing covariant derivatives of $R$ vanish. Using (\[Bachproperties\]), the trace of the field equations thus immediately implies R=4. \[R=4Lambda\] By substituting this relation into the field equations (\[GenQGFieldEq\]), they simplify considerably to R\_[ab]{}-g\_[ab]{}=4k B\_[ab]{}, k . \[fieldeqsgen\]

In this paper, *we restrict ourselves to* investigation of solutions with *vanishing cosmological constant* $\Lambda$ (see [@SvarcPodolskyPravdaPravdova:2018] for the study of a more general case ${\Lambda\ne0}$). In view of (\[R=4Lambda\]), this implies vanishing Ricci scalar, R=0, \[R=0\] and the field equations (\[fieldeqsgen\]) further reduce to a simpler form R\_[ab]{}=4k B\_[ab]{}, \[fieldeqsEWmod\] where the constant $k$ is now a shorthand for the combination of the theory parameters ${ k \equiv 