  The paper deals with the existence of standing wave solutions for the
Schr\"odinger-Poisson system with prescribed mass in dimension $N=2$. This
leads to investigate the existence of normalized solutions for an
integro-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 a
given 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 of
the unknowns. With respect to the related higher dimensional cases, the
presence of the logarithmic kernel, which is unbounded from above and below,
makes the structure of the solution set much richer and it forces the
implementation of new ideas to catch the normalized solutions.
