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We study the evolution of an open quantum system described by a dynamicalsemigroup having the Lindblad superoperator as a generator. This generator mayhave an eigenfunction with a unity eigenvalue, referred to as a constant ofmotion (COM). An open quantum system has a unique stationary state if and onlyif it has no COMs. A system with multiple stationary states has a basis ofCOMs; any COM of the system is a linear combination of the basis COMs. Thebasis divides the space of system states into subspaces. Each subspace has itsown stationary state, and any stationary state of the system is a linearcombination of these states. Usually, neither the basis of COMs nor even thenumber of COMs is known. We demonstrate that finding the stationary state ofthe system does not require looking for the COMs. Instead, one can construct aset of invariant subspaces. If the system evolution begins from one of thesesubspaces, the system will remain in it, arriving at a stationary stateindependent of evolution in other subspaces. We suggest a direct way of findingthe invariant subspaces by studying the evolution of the system. We show thatthe sets of invariant subspaces and subspaces generated by the basis of COMsare equivalent. A stationary state of the system is a weighted sum ofstationary states in each invariant subspace; the weighted factors aredetermined by the initial state of the system.

Zeroth law of thermodynamics for thermalized open quantum systems having integrals of motion