Non-vacuum twisting type-\(N\) metrics (Q2704185)

For the Bach equation the authors do not cite the original source, here it is: \textit{R. Bach}, Math. Z. 9, 110-135 (1921; JFM 48.1035.01): ``Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs.'' Every space-time conformally related to an Einstein space is a solution of the Bach equation; this is a trivial statement. The inversion is not at all trivial, and in Section 6 the authors give an example of a metric which solves the Bach equation but is not conformal to an Einstein space. Their last sentence reads: ``To the best of our knowledge this is the only known example\dots''. However, earlier papers to give such example already exist: in [\textit{H.-J. Schmidt}, ``Nontrivial solutions of the Bach equation exist'', Ann. Phys. (Leipz.), 41, 435-436 (1984)], such an example (explicit up to a quadrature) was given, and in [\textit{J. Demaret}, \textit{L. Querella} and \textit{C. Scheen}, ``Hamiltonian formulation and exact solutions of the Bianchi type I spacetime in conformal gravity'', Classical Quantum Gravity 16, 749-768 (1999; Zbl 0946.83049)] a fully explicit example was given.