Characterization of Symmetry Properties of First Integrals for Submaximal Linearizable Third-Order ODEs

الملخص EN

The relationship between first integrals of submaximal linearizable third-order ordinary differential equations (ODEs) and their symmetries is investigated.

We obtain the classifying relations between the symmetries and the first integral for submaximal cases of linear third-order ODEs.

It is known that the maximum Lie algebra of the first integral is achieved for the simplest equation and is four-dimensional.

We show that for the other two classes they are not unique.

We also obtain counting theorems of the symmetry properties of the first integrals for these classes of linear third-order ODEs.

For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2, and 3 symmetries, and for the 4 symmetry class of linear third-order ODEs, they are 0, 1, and 2 symmetries, respectively.

In the case of submaximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals.