  This paper is concerned with the following planar Schr\"{o}dinger-Poisson
system \begin{equation*} \begin{cases}
  -\triangle{u}+V(x)u+\phi{(x)}|u|^{p-2}u=f(x,u),&\text{in $\mathbb{R}^{2}$},
  \triangle{\phi}=|u|^{p},&\text{in $\mathbb{R}^{2}$}, \end{cases}
\end{equation*} where $p\geq2$ is a constant, $V(x)$ and $f(x,t)$ are
continuous, mirror symmetric or rotationally periodic functions. By assuming
that the nonlinearity $f(x,t)$ has critical exponential growth, we obtain a
nontrivial solution or a ground state solution of Nehari type to the above
system. Our results extend previous works of Cao_Dai_Zhang and Chen-Tang. We
handle more general nonlinearities $f$ with weaken constraint at infinity, and
we assume only the (AR) type condition to take place of the monotonicity
assumption. We considered all the cases $p\geq2$, and we show the existence of
solutions with multiple types of symmetry. As in Chen_Tang, we adopt a version
of mountain pass structure which provides a Cerami sequence, with two
innovative points. First, we make a key observation for the sign of a crucial
part of the energy functional corresponding to the nonlocal term
$\phi|u|^{p-2}u$, and secondly we adopt a new Moser type functions to ensure
the boundedness and compactness of the Cerami sequence. Moreover, our approach
works also for the subcritical growth case, and generalizes recent works
Liu_Radulescu_Tang_Zhang,Cao_Dai_Zhang,Chen_Tang.
