Algebraic Properties of First Integrals for Scalar Linear Third‐Order ODEs of Maximal Symmetry

By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third‐order ordinary differential equations (ODEs) and their point symmetries. It is well known that there are three classes of linear third‐order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third‐order equation has seven‐point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y′′′ = 0 is unique and four‐dimensional. Moreover, we show that the Lie algebra of the simplest linear third‐order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third‐order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher‐order ODEs of maximal symmetry from two of their basic integrals.