Halphen's Transform for the Lamé Equation and Ermakov–Painlevé XXV Equation

In S. Reiter (J. reine angew. Math. 704, 2015) it was shown that the Halphen transform of a Lamé equation can be written as the symmetric square of this equation followed by an Euler transformation via third‐order linear differential equations. In this paper, we examine nonlinear second‐order differential equations, which are associated to the third‐order linear differential equations via the Schwarzian derivative, and determine whether they are reduced to either the Ermakov equation or to the Painlevé XXV equation.