Symmetry and singularity properties of second-order ordinary differential equations of Lie's type III

We extend the work of Abraham-Shrauner [B. Abraham-Shrauner, Hidden symmetries and linearization of the modified Painlevé–Ince equation, J. Math. Phys. 34 (1993) 4809–4816] on the linearization of the modified Painlevé–Ince equation to a wider class of nonlinear second-order ordinary differential equations invariant under the symmetries of time translation and self-similarity. In the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations.