Equivalence classes of the second order ODEs with the constant Cartan invariant

TL;DR

This paper classifies second order ODEs with constant invariants, identifying four main types and providing comprehensive lists of nonequivalent equations, including examples like Painleve and Emden equations.

Contribution

It introduces a complete classification of second order ODEs with constant invariants and solves the equivalence problem for several notable equations.

Findings

Identified four basic types of second order ODEs with constant invariants.

Provided complete lists of nonequivalent equations for each type.

Solved the equivalence problem for Painleve II, Painleve III, and Emden equations.

Abstract

Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the exampes the equivalence problem for the Painleve II equation, Painleve III equation with three zero parameters, Emden equations and for some other equations is solved.