Solutions of a modified third Painlevé equation are meromorphic
It is known that all solutions to the first, second and fourth Painlevé differential equations are meromorphic in the plane. That is in general not true for the third Painlevé equation \[ w''= {(w')^2\over w}-{w' \over z}+{\alpha w^2+\beta \over z}+\gamma w^3+{\delta\over w}, \quad \alpha, \beta,\gamma, \delta\in \mathbb{C}. \] Using the transformation \(W(\zeta)= w(z)z\) and \(z=e^{\zeta/2}\) one gets the modified equation \[ W''={(W')^2\over W}+\alpha W^2+ \gamma W^3+\beta e^{\zeta}+ {\delta e^{2\zeta}\over W} \] with \((4\alpha,4 \beta, 4\gamma,4 \delta)\) instead of \((\alpha,\beta, \gamma,\delta)\). Now the authors prove that all solutions to this modified equation again are meromorphic functions in the plane.
scientific article; zbMATH DE number 1765851