Lie algebras associated with scalar second-order ordinary differential equations

Second‐order ordinary differential equations are classified according to their Lie algebra of point symmetries. The existence of these symmetries provides a way to solve the equations or to transform them to simpler forms. Canonical forms of generators for equations with three‐point symmetries are established. It is further shown that an equation cannot have exactly r ∈{4,5,6,7} point symmetries. Representative(s) of equivalence class(es) of equations possessing s ∈{1,2,3,8} point symmetry generator(s) are then obtained.