We will study a new special solution of the fourth Painlevé equation, for which we can calculate the linear monodromy exactly. We will show the relation between Umemura's classical solutions and our solutions.
Kazuo Kaneko. "A new solution of the fourth Painlevé equation with a solvable monodromy." Proc. Japan Acad. Ser. A Math. Sci. 81 (5) 75 - 79, May 2005. https://doi.org/10.3792/pjaa.81.75