The paper addresses the experimental technique that, when applied to a 2D-electron system in the integer quantum Hall regime with filling factor <$E nu~=~2> (the Hall insulating state), allows resonant excitation of magnetoexcitons, their detection, control of an ensemble of long-lived triplet excitons and investigation of their radiationless decay related to exciton spin relaxation into the ground state. The technique proposed enables independent control of photoexcited electrons and Fermi-holes using photoinduced resonance reflection spectra as well as estimate with a reasonable degree of accuracy the resulting density of photoinduced electron-hole pairs bound into magnetoexcitons. The mere existence of triplet excitons was directly established by inelastic light scattering spectra which were analyzed to determine the value of singlet-triplet exciton splitting. It was found that the lifetimes of triplet excitons conditioned by electron spin relaxation in highly perfect GaAs/AlGaAs heterostructures with highly mobile 2D electrons are extremely long exceeding 100 <$Emu>s at <$ET~<<~1> K. The paper presents a qualitative explanation of the long-spin relaxation lifetimes which are unprecedented for translation-invariant triplet magnetoexcitons, electrically neutral excitations following Bose - Einstein statistics, in a Fermi electron system and investigate their collective properties. At sufficiently high densities of triplet magnetoexcitons and low temperatures, <$ET~<<~1> K, the degenerate magnetofermionic system exhibits condensation of the triplet magnetoexcitons into a qualitatively new collective state with unusual properties which occurs in the space of generalized moments (magnetic translation vectors). The occurrence of a condensed phase is accompanied with a significant decrease in the viscosity of the photoexcited system, which is responsible for electron spin transport at macroscopic distances, as well as with the effects of threshold enhancement of the system response to the external action of the electromagnetic field and emergence of a new intensive radiative recombination channel.

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