We study the normalized solutions of the \(L^2\) -critical Schrödinger–Poisson system with an external potential \(V(x)=|x|^2\) in \({\mathbb {R}}^2\) , which can be described by the constraint minimization problem. When the magnetic field is attractive, we prove that there is a threshold \(a^*\in (0,\infty )\) such that the constraint minimizer exists if and only if the interaction strength \(a<a^*\) . Moreover, for the repulsive case, there exists a minimizer if \(a<a^*\) , while there does not exist any minimizer if \(a>a^*\) . Particularly, after analyzing its limiting behavior, we then obtain the uniqueness of positive minimizers as \(a\nearrow a^*\) by overcoming the sign-changing property of the logarithmic convolution and the non-invariance under translations of the harmonic potential.
