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

A protocol for explicitly constructing the exact time-evolution operators generated by $2 times 2$ time-dependent $PT$-symmetry Hamiltonians is reported. Its mathematical applicability is illustrated with the help of appropriate examples. The physical relevance of the proposed approach within gain-loss system scenarios, like two-coupled wave-guides, is discussed in detail.