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The paper deals with the existence of standing wave solutions for theSchr\"odinger-Poisson system with prescribed mass in dimension $N=2$. Thisleads to investigate the existence of normalized solutions for anintegro-differential equation involving a logarithmic convolution potential,namely $$ \left \{ \begin{aligned} - \Delta u & + \lambda u + \gamma \Bigl(\log{| \cdot |} * |u|^2 \Bigr) u =a |u|^{p-2} u \qquad \text{in $\mathbb R^2$,} \\&\int_{\mathbb R^2} |u|^2 dx = c \end{aligned} \right. $$ where $c>0$ is agiven real number. Under different assumptions on $\gamma \in \mathbb R$, $a\in \mathbb R$, $p>2$, we prove several existence and multiplicity results.Here $\lambda \in \mathbb R $ appears as a Lagrange parameter and is part ofthe unknowns. With respect to the related higher dimensional cases, thepresence of the logarithmic kernel, which is unbounded from above and below,makes the structure of the solution set much richer and it forces theimplementation of new ideas to catch the normalized solutions.

Stationary waves with prescribed $L^2$-norm for the planar Schr\"odinger-Poisson system