Abstract: The paper deals with the following planar Schrödinger-Poisson system $\begin{equation*} \begin{cases}-\Delta u+\lambda u+\phi u= f(u),&\text{in}~\mathbb{R}^{2},\\-\Delta\phi= u^2,&\text{in}~\mathbb{R}^{2},\\ \int_{\mathbb{R}^{2}}u^2{\rm d}x=a^2,~u\in H^1(\mathbb{R}^2), \end{cases} \end{equation*}$ where $a\in(0,1)$, $\lambda \in \mathbb{R}$ is an undetermined parameter which appears as a Lagrange multiplier and $f$ satisfies the exponential critical growth. Under suitable conditions on $f$, we manage to establish such critical points which emerge as a local minimizer or correspond to a mountain pass. Furthermore, by using genus theory, we obtain infinitely many solutions. The proofs are based upon the reduction method by working on a natural constraint, which is introduced by Bartsch and Soave [J Funct Anal, 2017, 272: 4998-5037]. Our results complement the results made by Cingolani and Jeanjean [SIAM J Math Anal, 2019, 51: 3533-3568] where handle the Sobolev subcritical case.

Key words: Schrö, dinger-Poisson system, normalized solution, exponential critical growth, variational methods