First integrals, integrating factors and λ-symmetries of second-order differential equations

For a given second-order ordinary differential equation (ODE), several relationships among first integrals, integrating factors and lambda-symmetries are studied. The knowledge of a lambda-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. If two nonequivalent lambda-symmetries of the equation are known, then an algorithm to find two functionally independent first integrals is provided. These methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or in the Prelle-Singer method. These results are applied to several ODEs that appear in the study of relevant equations of mathematical physics.