Free space strange and unipolar EM pulses: yes or no?

E.G. Bessonov suggested the time integrated strength of electric field $\int_{-\infty}^{\infty}E(r,t)dt=S_E(r)$ as a parameter to classify electromagnetic (EM) waves. Since then, this parameter has been studied and used in many works on microwave and laser physics, especially when it comes to unipolar, bipolar and few cycle EM pulses. In this paper, it is shown that $S_E(r)=0$ is an identity for a wide class of free space pulses of finite total energy. This property can be useful in various applications of few cycle radiation and as a benchmark in EM and QED computations.