Integration of ordinary differential equation with a small parameter via approximate symmetries: Reduction of approximate symmetry algebra to a canonical form

Method of integration of second-order ordinary differential equations with twodimensional Lie symmetry algebras by reducing basic symmetries to canonical forms is extended to second-order equations with a small parameter for their approximate integration using two essential approximate symmetries. Canonical forms of basic operators of corresponding approximate Lie algebras L r , r = 2, 3, 4, as well as general forms of invariant differential equations and their solutions are presented. The similar problems are also solved for systems of two first-order ordinary differential equations with two approximate symmetries.