Integrals of motion of pure and mixed quantum systems

We analyse and compare the properties of the integrals of motion of two types of quantum systems: pure systems which can be described in terms of the wave functions obeying the Schrödinger equation and mixed systems described in terms of density matrix. The model equation for the Wigner function of the damped quantum oscillator is considered as an example. It is shown that for dissipative non-Hamiltonian quantum systems arbitrary functions of conserved quantities in the general case are not conserved quantities. The property that any function of integrals of motion is also an integral of motion, is valid only for the systems with Hamiltonians.