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After the discovery of the Higgs-like boson with the mass 125 GeV at the LHC in 2012 a lot of effort has been applied to the investigation of its properties and an identification of the Higgs-like particle as the Standard Model Higgs boson, Higgs boson in the general extensions of the Standard Model by a higher-dimension effective operators in the Higgs, Yukawa and gauge sectors, or an identification as the Higgs boson of the minimal supersymmetric standard model in the decoupling limit. A survey of the identifiation results is going to be presented consistenlty with the recent data on the astrophysical aspects, connected with the existence of the Higgs system with the low-lying state at the mass 125.02 GeV.