**4. Conclusions**

In this paper, we have applied the complex Noether approach and attempted to classify a two-dimensional coupled system of LE equations that appears in physics and applied mathematics with respect to Noether-like-operators and corresponding first integrals by taking the functions *F*1 and *F*2 in their more general forms in Equation (11). In this study, we have observed that for some of the systems of LE equations, every pair of Noether-like operators forms an Abelian Lie algebra. We have also highlighted that for certain pairs of Lagrangians, the Noether-like operators become Noether symmetries of the Euler–Lagrange systems of LE equations and give rise to the same Noetherian first integrals as we determined from our complex approach. Therefore, the study of invariant quantities of many dynamical systems can be made with the help of complex Lagrangian formalism, which seems to be more simple and elegant.

**Funding:** This research received no external funding.

**Acknowledgments:** M.U. Farooq is thankful to Fazal M. Mahomed for his important comments on this paper. The author also expresses his gratitude to the referees for some useful suggestions.

**Conflicts of Interest:** The author declares no conflict of interest.
