  We look for solutions to the Schr\"{o}dinger-Poisson-Slater equation $$-
\Delta u + \lambda u - \gamma (|x|^{-1} * |u|^2) u - a |u|^{p-2}u = 0 \quad
\mbox{in} \quad \mathbb{R}^3 \quad \mbox{which satisfy} \quad
||u||_{L^2(\mathbb{R}^3)}^2 = c $$
  for some prescribed $c>0$. Here $ u \in H^1(\mathbb{R}^3)$, $\gamma \in
\mathbb{R},$ $ a \in \mathbb{R}$ and $p \in (\frac{10}{3}, 6]$. Both in the
Sobolev subcritical case $p \in (\frac{10}{3}, 6)$ and in the Sobolev critical
case $p=6$, we show that there exists a $c_1>0$ such that, for any $c \in
(0,c_1)$, this equation admit two solutions $u_c^+$ and $u_c^-$ which can be
characterized respectively as a local minima and as a mountain pass critical
point of the associated Energy functional restricted to the norm constraint.
The behavior, as $c \to 0$, of the associated Lagrange multipliers is
described.
