On the symmetry solutions of two-dimensional systems not solvable by standard symmetry analysis

TL;DR

This paper identifies a class of 2D second-order ODE systems that require fewer symmetries for solutions, introducing a geometric approach to distinguish linearizable and solvable systems.

Contribution

It introduces a novel classification method for 2D second-order ODE systems based on symmetry requirements and geometric criteria.

Findings

Identifies systems needing fewer symmetries than standard methods

Provides a diagrammatic geometric framework for linearizability

Distinguishes linearizable, complex-linearizable, and solvable systems

Abstract

A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are linearizable, complex-linearizable and solvable systems. We also present the underlying concept diagrammatically that provides an analogue in $\Re^{3}$ of the geometric linearizability criteria in $\Re^2$.