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The nonlinear pseudo-oscillator recently tackled by Gadella and Lara ismapped to an Emden-Fowler (EF) equation that is written as an autonomoustwo-dimensional ODE system for which we provide the phase-space analysis andthe parametric solution. Through an invariant transformation we find periodicsolutions to a certain class of EF equations that pass an integrabilitycondition. We show that this condition is necessary to have periodic solutionsand via the ODE analysis we also find the sufficient condition for periodicorbits. EF equations that do not pass integrability conditions can be madeintegrable via an invariant transformation which also allows us to constructperiodic solutions to them. Two other nonlinear equations, a zero-frequencyErmakov equation and a positive power Emden-Fowler equation are discussed inthe same context

Existence of periodic orbits in nonlinear oscillators of Emden-Fowler form