The power-law expansion of the general solution of the Chew–Low equations [3] proposed by the authors [1, 2] is considered in the neighborhood of the point $w=0$. It is shown that, in contrast to the quadratic approximation, the cubic approximation does not have the required Born pole at this point. It is concluded from this that the expansion is not valid near the Born pole. In the class of physically interesting solutions, the local restrictions $\beta(0)=0$ and $C(0)\ne0$ are obtained for the arbitrary periodic functions $\beta(w)$ and $C(w)$ that determine the general solution. By numerical analysis, the value $C(0)\approx-265$ is obtained for solutions with Born pole.