In this note, we will completely classify all of rational solutions of the $A_4$ Painlevé equation, which is a generalization of the fourth Painlevé equation. The rational solutions are classified to three types by the Bäcklund transformation group.
Kazuhide Matsuda. "Rational solutions of the $A_4$ Painlevé equation." Proc. Japan Acad. Ser. A Math. Sci. 81 (5) 85 - 88, May 2005. https://doi.org/10.3792/pjaa.81.85