Meromorphic Painlevé transcendents at a fixed singularity (Q2841689)

This paper is an extension of the study of the authors on the Painlevé equations which have fixed singular points of the Briot-Bouquet type [\textit{K. Kaneko}, Proc. Japan Acad., Ser. A 82, No. 5, 71--76 (2006; Zbl 1134.33324); \textit{K. Kaneko} and \textit{Y. Ohyama}, Funkc. Ekvacioj, Ser. Int. 50, No. 2, 187--212 (2007; Zbl 1162.34074)]. In particular, applying Briot-Bouquet's theorem to the Hamiltonian systems corresponding to \(P'_{III}(D^{(1)}_6),\,P'_{III} (D^{(1)}_7)\) and \(P'_{III}(D^{(1)}_8)\) equations, they find all of the meromorphic solutions around the origin of these equations. The authors also compute the monodromy data of the isomonodromy deformation equations corresponding to the third, fifth and sixth Painlevé equations taken at the meromorphic solutions around the origin.