Initial states of qubit–environment models leading to conserved quantities

It is possible to prepare a composite qubit–environment system so that its time evolution will guarantee the conservation of a preselected qubit's observable. In general, this observable is not associated with a symmetry. The latter may not even be present in the subsystem. The initial states which lead to such a quantity conserved dynamics form a subspace of the qubit–environment space of states. General construction of this subspace is presented and illustrated by two examples. The first one is the exactly solvable Jaynes–Cummings model and the second is the multi-photon Rabi model. (paper)