Symmetry classification of first integrals for scalar dynamical equations

We completely classify the first integrals of scalar non-linear second-order ordinary differential equations (ODEs) in terms of their Lie point symmetries. This is performed by first obtaining the classifying relations between point symmetries and first integrals of scalar non-linear second-order equations which admit one, two and three point symmetries. We show that the maximum number of symmetries admitted by any first integral of a scalar second-order non-linear ODE is one which in turn provides reduction to quadratures of the underlying dynamical equation. We provide physical examples of the generalized Emden–Fowler, Lane–Emden and modified Emden equations.