First integrals for a generalized coupled Lane-Emden system (Q619744)

The generalized coupled Lane-Emden system \[ \frac{d^2u}{dt^2}+\frac{n}{t}\frac{du}{dt}+f(t)v^q=0,\quad \frac{d^2v}{dt^2}+\frac{n}{t}\frac{dv}{dt}+f(t)u^p=0 \] is considered, where \(n,p,q\) are real constants and \(f\) is an arbitrary real-valued function. The authors study the complete Noether symmetry classification of this system with respect to the standard first-order Lagrangian. Several cases for the function \(f\) which result in Noether point symmetries are obtained. For each case, the authors obtain a first integral for the corresponding Noether operator.