**M. Umar Farooq**

Department of Basic Sciences and Humanities, College of E&ME, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan; m\_ufarooq@yahoo.com

Received: 18 December 2018; Accepted: 15 January 2019; Published: 1 February 2019

**Abstract:** Coupled systems of Lane–Emden equations are of considerable interest as they model several physical phenomena, for instance population evolution, pattern formation, and chemical reactions. Assuming a complex variational structure, we classify the generalized system of Lane–Emden type equations in relation to Noether-like operators and associated first integrals. Various forms of functions appearing in the considered system are taken, and it is observed that the Noether-like operators form an Abelian algebra for the corresponding Euler–Lagrange-type systems. Interestingly, we find that in many cases, the Noether-like operators satisfy the classical Noether symmetry condition and become the Noether symmetries. Moreover, we observe that the classical Noetherian integrals and the first integrals we determine using the complex Lagrangian approach turn out to be the same for the underlying system of Lane–Emden equations.

**Keywords:** generalized Lane–Emden systems; Noether-like operator; conservation laws
