Integration of Systems of ODEs via Nonlocal Symmetry-Like Operators

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex plane. Examples are taken from the literature. Precisely it is shown how the reduction to quadratures of a system of two second-order ordinary differential equations that admits four Lie-like operators with certain structure is obtainable from a restricted complex ordinary differential equation possessing a connected two-dimensional complex Lie algebra. A direct method of integration for a system of two first and second-order equations which possess nonlocal symmetry-like operators are given. Moreover, we present the use of nonlocal Noether-like operators to effect double reduction of order of systems of two second-order equations that arise from the corresponding scalar complex Euler- Lagrange equations which admit nonlocal Noether symmetry.