Yes, the literature on this is quite sparse, and mostly poor.In response to this I was searching for information about fractional linear transforms. All of them that I saw were mappings from the complex plane to the complex plane. I didn’t see anything on FL transforms as a mapping from R4 to R4.
In Fock & Kemmer, [Ref: FK64, Appendix A] there's a derivation of the most general transformations that map solutions of the free EoMs among themselves.I am not sure how they are even applicable here.
Stepanov [Ref: Step99, Appendix 1] gives a simplified derivation in 1+1D (although the main body of that paper is rather poor, IMHO).
These transformations are also known as "Fock-Lorentz" transformations (which coincidenally has the same initials "FL"). But you can mostly ignore the Wikipedia page on that subject, since it gives an impression that FL transformations necessarily involve a varying (local) speed of light, which is a false claim.
Kerner [Ref: Ker76] also attempted some work on this, but he didn't get very far and (in subsequent publications) develops an increasingly aggressive/desperate tone. He progresses to de Sitter, but doesn't get very far beyond that.
Manida [Ref: Man99], also derives Fock-Lorentz transformations, duplicating some of Kerner's early work (though apparently without citing him). But his attempts to develop this into a cosmoglogy are (imho) fruitless, with shortcomings reminiscent of Milne's work.
FK64: V. Fock, N. Kemmer (translator), The theory of space, time and gravitation.
2nd revised edition. Pergamon Press, Oxford, London, New York, Paris (1964).
Step99: S. S. Stepanov,
Fundamental Physical Constants & the Principle of Parametric Incompleteness,
Ker76: E. H. Kerner,
An extension of the concept of inertial frame and of Lorentz transformation,
Proc. Nat. Acad. Sci. USA, Vol. 73, No. 5, pp. 1418-1421, May 1976 .
Man99: S. N. Manida,
Fock-Lorentz transformations and time-varying speed of light,
Available as: arXiv:gr-qc/9905046 .
(Ignore the 2nd part of the title: he's not talking about a varying local speed of light.)