Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians

For a large class of time-dependent non-Hermitain Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis-Riesenfeld invariants making useof the metric picture, which is an equivalent alternative to the Schr\"{o}dinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian.