The radio physical oscillator model is a nonlinear three-dimensional autonomous system of first-order ordinary differential equations with quintic polynomial. This model explains a generator of hard oscillations with coupled to a rechargeable power source and depending on four non negative parameters. The present work is divided into two parts: the first part investigates the Darboux integrability of this system. More particularly, we have determined all the invariant algebraic surfaces as well as all the exponential factors of this system. For any value of parameters, we verify that the radio physical oscillator system does not admit polynomial, Darboux and rational first integrals through the analysis of its Darboux polynomials and its exponential factors, while the second part focuses on the investigation of the periodic solutions that may bifurcate from the zero-Hopf equilibrium points using first-order averaging theory, but unfortunately, the method is unable to provide any information about the periodic solution of the system.