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The Noether-like operators that play an essential role in writing down theinvariants for systems of two ordinary differential equations (ODEs) areconstructed. The classification of such operators is carried out with the helpof analytic continuation of the Lagrangians on the line. Cases in which theNoether-like operators are also Noether symmetries for the systems of ODEs arebriefly mentioned. In particular, the 8-dimensional maximal Noether subalgebrais remarkabely obtained for the simplest system of the free particle equationsin two dimensions from the 5-dimensional complex Noether algebra. We presentthe effectivness of Noether-like operators as well as the determination of allfirst integrals of systems of nonlinear differential equation which have notbeen reported before. This study gives a new direction to construct firstintegrals for systems of nonlinear differential equations.

Two-dimensional Systems that Arise from the Noether Classification of Lagrangians on the Line