Lie symmetry analysis is one of the powerful tools to analyse nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, 𝜆-symmetries, adjoint symmetries and telescopic vector fields of a secondorder ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between
symmetries and integrating factors and
symmetries and integrals are also discussed and illustrated through the same example.
The interconnections between some of the above symmetries, i.e.,
Lie point symmetries and 𝜆-symmetries and
exponential nonlocal symmetries and 𝜆-symmetries are also discussed.
The order reduction procedure is invoked to derive the general solution of the second-order equation.
