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A protocol for explicitly constructing the exact time-evolution operatorsgenerated by $2 \times 2$ time-dependent $PT$-symmetry Hamiltonians isreported. Its mathematical applicability is illustrated with the help ofappropriate examples. The physical relevance of the proposed approach withingain-loss system scenarios, like two-coupled wave-guides, is discussed indetail.

Analytically solvable $2\times2$ $PT$-symmetry dynamics from su(1,1)-symmetry problems