Characterization of several classes of second-order ODEs by using differential invariants (Q2368488)

Lie presented a complete group classification of scalar second-order ODEs in the complex domain. He also gave an invariant criterion for linearization, via invertible transformations, for such equations. Later, Tresse, a student of Lie, pursued invariants of continuous groups and reductions of second-order ODEs. Since then several authors have studied second-order ODEs using transformations including the Cartan equivalence method. In this work, the authors use invariants with respect to subgroups of horizontal, i.e., changes of independent variables, and vertical (changes of dependent variables) transformations. They calculate invariants and obtain conditions for reduction of five classes of second-order ODEs, including three important Painlevé equations and the linear equations' class. This approach provides a way, via invariants, to check when a given second-order ODE belongs to a canonical form under horizontal or vertical subgroups of transformations. An appendix is given for the computational aspects for which \texttt{MATHEMATICA} is used. The paper has interest. However, the reference list misses some important works on the complete group classification of scalar second-order ODEs in the real domain.