Integration of systems of ordinary differential equations with a small parameter which admit approximate Lie algebras

The algorithm for the order reduction of ordinary differential equations (ODEs) by using the operator of invariant differentiation (OID) of admitted Lie algebra is modified for systems of ODEs with a small parameter that admit approximate Lie algebras of operators. Invariant representations of second-order ODEs and systems of two second-order ODEs are presented. The OID of approximate Lie algebra is introduced. It is shown that it is possible to construct a special type of OID, which is used for obtaining the first integral of the system considered. Examples of using the algorithm for cases of complete and incomplete inheritance of a Lie algebra are given.