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In this paper, we investigate a class of planar Schr\"{o}dinger-Poissonsystems with critical exponential growth. We establish conditions for the localwell-posedness of the Cauchy problem in the energy space, which seemsinnovative as it was not discussed at all in any previous results. Byintroducing some new ideas and relaxing some of the classical growthassumptions on the nonlinearity, we show that such system has at least twostanding waves with prescribed mass, where one is a ground state standing waveswith positive energy, and the other one is a high-energy standing waves withpositive energy. In addition, with the help of the local well-posedness, weshow that the set of ground state standing waves is orbitally stable.

The planar Schrodinger--Poisson system with exponential critical growth: The local well-posedness and standing waves with prescribed mass