  We investigate the existence, uniqueness, and radial symmetry of normalized
solutions to the Schr\"{o}dinger Poisson equation with non-autonomous
nonlinearity $f(x,u)$: \begin{equation} -\triangle
u+(|x|^{-1}*|u|^2)u=f(x,u)+\lambda u, \nonumber \end{equation} subject to the
constraint $\mathcal{S}_c=\{u\in H^1(\mathbb{R}^3)|\int_{\mathbb{R}^3}u^2=c>0
\}$. We consider three cases based on the behavior of $f(x,u)$: the $L^2$
supercritical case, the $L^2$ subcritical case with growth speed less than
three power times, and the $L^2$ subcritical case with growth speed more than
three power times. We establish the existence of solutions using three
different methods depending on $f(x,u)$. Furthermore, we demonstrate the
uniqueness and radial symmetry of normalized solutions using an implicit
function framework when $c$ is small.
