I have a vectorial function of the electric field $\overline E$ of a charge with velocity $\overline{v}$ (along the axis $x$). This function $\overline E$ (or E), is shown in the following figure,
$$\overline E=\overline E_r(r, \vartheta,0)=k_e\,\dfrac q{r^2}\,\hat{\textbf r}\,\dfrac{1-\beta^2}{(1-\beta^2\,\sin^2\vartheta)^{3/2}}$$ with $\overline E_r(r, \vartheta,0)=\overline E_r(r, \vartheta,t)$ and $(t=0)$. where $r=d(q,P)$ at $(t=0)$, with $r$ the distance between the charge and a point $P\in \mathbb{R}^3$, $\vartheta$ is the angle between the axis $x$ and $\overline{r}$ with $\overline{E}$ that have the same direction of $\overline{r}$.
Considering the spherical coordinate (polar coordinate in the space), why the components of the electric field $\overline E$ are independent to azimuthal coordinate $\varphi$ (horizontal angle), $E_\varphi=0$, and of the colatitude $\vartheta$, $E_\vartheta=0$?