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The spontaneous breaking of the continuous symmetries of thetwo-dimensional(2D) electron-hole systems in a strong perpendicular magneticfield leads to the formation of new ground states and determines the energyspectra of the collective elementary excitations appearing over these groundstates. In this review the main attention is given to the electron-hole systemsforming coplanar magnetoexcitons in the Bose-Einstein condensation(BEC) groundstate with the wave vector k=0, taking into account the excited Landau levels,when the exciton-type elementary excitations coexist with the plasmon-typeoscillations. At the same time properties of the two-dimensional electrongas(2DEG) spatially separated as in the case of double quantum wells(DQWs) fromthe 2D hole gas under conditions of the fractional quantum Hall effect(FQHE)are of great interest because they can influence the quantum states of thecoplanar magnetoexcitons when the distance between the DQW layers diminishes.We also consider in this review the bilayer electron systems under conditionsof the FQHE with the one half filling factor for each layer and with the totalfilling factor for two layers equal to unity because the coherence between theelectron states in two layers is equivalent to the formation of the quantumHall excitons(QHExs) in a coherent macroscopic state. The breaking of theglobal gauge symmetry as well as of the continuous rotational symmetries leadsto the formation of the gapless Nambu-Goldstone(NG) modes while the breaking ofthe local gauge symmetry gives rise to the Higgs phenomenon characterized bythe gapped branches of the energy spectrum. The conditions in which thespontaneous coherence could appear in a system of indirect excitons in a doublequantum well structures are discussed. The experimental attempts to achievethese conditions, the main results and the accumulated knowledge are reviewed.

Spontaneous symmetry breaking and coherence in two-dimensional electron-hole and exciton systems