The λ -symmetry approach is applied to a family of second-order ODEs whose algebra of Lie point symmetries is insufficient to integrate them.The general solution and two functionally independent first integrals of a subclass of the studied equations can be expressed in terms of a fundamental set of solutions of a related second-order linear ordinary differential equation.Remarkably, no equation in the family passes the Lie's test of linearisation, although all of them can be linearised by generalised Sundman transformations.Several examples, including equations lacking Lie point symmetries, illustrate the presented procedure.