We investigate the effect of the electric field on the existence and propagation of axisymmetric solitary waves in fluid-filled thin tubes by direct analysis of the governing field equations. For a specific wave speed, the solution of the amplitude only requires to find the roots of an algebraic equation. In our search for solitary waves, we made use of a specific material model, namely the electroelastic neo-Hookean material, for a thin-walled tube, which is charged through a constant difference of potential through its soft flexible electrodes on the inside and outside surfaces. Then, we analyze the effects of the electric field, which appears as a result of the difference of potential on the emergence of the solitary wave. When searching for the domain of solitary waves under given longitudinal and circumferential prestretch conditions, we found a narrow set of combinations of the fluid velocity and the wave speed that give rise to solitary waves. This domain is bounded by several curves which represent various physical and mathematical restrictions. Surprisingly, these restrictions can be placed as purely algebraic conditions, which are applied to the governing system of differential equations. Foremost among the physical restrictions are the avoidance of wrinkles and the self-impenetrability of the wave profile. In particular, the existence of a critical wave speed of imminent wrinkling, independent of the background fluid velocity, is established carefully. Once we consider these and other criteria, we discuss the possible existence of supercritical and retrograde waves.