  In this paper, we investigate the normal weighed composition operators
$W_{\psi,\varphi}$ which is $\mathcal{J}-$symmetric, $\mathcal{C}_1-$symmetric
and $\mathcal{C}_2-$symmetric on the Hardy space $H^2(\mathbb{D})$
respectively. Firstly, equivalent conditions of the normality of
$\mathcal{C}_1-$symmetric and $\mathcal{C}_2-$symmetric weighted composition
operators on $H^2(\mathbb{D})$ is given. Furthermore, the normal
$\mathcal{J}-$symmetric, $\mathcal{C}_1-$symmetric and
$\mathcal{C}_2-$symmetric weighted composition operators on $H^2(\mathbb{D})$
when $\varphi$ has an interior fixed point, $\varphi$ is of hyperbolic type or
parabolic type are respectively investigated.
