All meromorphic solutions of the autonomous Schwarzian differential equations

This paper considers a specific autonomous Schwarzian differential equation given by and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper ( Math. Z . 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case.