  In this paper, we investigate the existence and local uniqueness of
normalized peak solutions for a Schr\"odinger-Newton system under the
assumption that the trapping potential is degenerate and has non-isolated
critical points.
  First we investigate the existence and local uniqueness of normalized
single-peak solutions for the Schr\"odinger-Newton system. Precisely, we give
the precise description of the chemical potential $\mu$ and the attractive
interaction $a$. Then we apply the finite dimensional reduction method to
obtain the existence of single-peak solutions. Furthermore, using various local
Pohozaev identities, blow-up analysis and the maximum principle, we prove the
local uniqueness of single-peak solutions by precise analysis of the
concentrated points and the Lagrange multiplier. Finally, we also prove the
nonexistence of multi-peak solutions for the Schr\"odinger-Newton system, which
is markedly different from the corresponding Schr\"odinger equation. The
nonlocal term results in this difference.
  The main difficulties come from the estimates on Lagrange multiplier, the
different degenerate rates along different directions at the critical point of
$P(x)$ and some complicated estimates involved by the nonlocal term. To our
best knowledge, it may be the first time to study the existence and local
uniqueness of solutions with prescribed $L^{2}$-norm for the
Schr\"odinger-Newton system.
