A note on autonomous Schwarzian differential equations

Abstract In this paper, we investigate a special autonomous Schwarzian differential equation This equation was overlooked by K. Ishizaki during his classification of autonomous differential equations with transcendental meromorphic solutions. It was discovered by Zhang–Liao–Wu–Zhao when they were exploring the rational solutions of autonomous differential equations. In fact, we prove this equation admits no meromorphic solutions, thus completing the classification of autonomous Schwarzian differential equations.