A relationship between λ ‐symmetries and first integrals for ordinary differential equations

In this paper, we provide some geometric properties of λ ‐symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of λ ‐symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries. We also investigate the properties of λ ‐symmetries in the sense of the deformed Lie derivative and differential operator. We show that λ ‐symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases.