  We provide exact analytical solutions for a two dimensional explicitly
time-dependent non-Hermitian quantum system. While the time-independent variant
of the model studied is in the broken PT-symmetric phase for the entire range
of the model parameters, and has therefore a partially complex energy
eigenspectrum, its time-dependent version has real energy expectation values at
all times. In our solution procedure we compare the two equivalent approaches
of directly solving the time-dependent Dyson equation with one employing the
Lewis-Riesenfeld method of invariants. We conclude that the latter approach
simplifies the solution procedure due to the fact that the invariants of the
non-Hermitian and Hermitian system are related to each other in a
pseudo-Hermitian fashion, which in turn does not hold for their corresponding
time-dependent Hamiltonians. Thus constructing invariants and subsequently
using the pseudo-Hermiticity relation between them allows to compute the Dyson
map and to solve the Dyson equation indirectly. In this way one can bypass to
solve nonlinear differential equations, such as the dissipative Ermakov-Pinney
equation emerging in our and many other systems.
