A one-parameter family of Emden–Fowler equations defined by Lampariello’s parameter p which, upon using Thomas–Fermi boundary conditions, turns into a set of generalized Thomas–Fermi equations comprising the standard Thomas–Fermi equation for p=1 is studied in this paper. The entire family is shown to be non integrable by reduction to the corresponding Abel equations whose invariants do not satisfy a known integrability condition. We also discuss the equivalent dynamical system of equations for the standard Thomas–Fermi equation and perform its phase-plane analysis. The results of the latter analysis are similar for the whole class.