Non-stationary SQM/IST Correspondence and $\mathcal{C}\mathcal{P}\mathcal{T}\mathrm{/}\mathcal{P}\mathcal{T}$-invariant paired Hamiltonians on the line

We fill some of existed gaps in the correspondence between Supersymmetric Quantum Mechanics and the Inverse Scattering Transform by extending the consideration to the case of paired stationary and non-stationary Hamiltonians. We formulate the corresponding to the case Goursat problem and explicitly construct the kernel of the non-local Inverse Scattering Transform, which solves it. As a result, we find the way of constructing non-hermitian Hamiltonians from the initially hermitian ones, that leads, in the case of real-valued spectra of both potentials, to pairing of $\mathcal{C}\mathcal{P}\mathcal{T}\mathcal{/}\mathcal{P}\mathcal{T}$-invariant Hamiltonians. The relevance of our proposal to Quantum Optics and optical waveguides technology, as well as to non-linear dynamics and Black Hole Physics is briefly discussed.