  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 soluble
Jaynes-Cummings model and the second is the multi-photon Rabi model.
