Universal integrals of motion and universal invariants of quantum systems

Universal quantum integrals of motion are introduced, and their relation wi th the universal quantum invariants is established. The invariants concerne d are certain combinations of the second- and higher-order moments (varianc es) of quantum-mechanical operators, which are preserved in time independen tly of the concrete form of the coefficients of the Schrodinger equation, p rovided the Hamiltonian is either a generic quadratic form of the coordinat e and momenta operators, or a linear combination of generators of some fini te-dimensional algebra (in particular, any semisimple Lie algebra). Using t he phase space representation of quantum mechanics in terms of the Wigner f unction, the relations between the quantum invariants and the classical uni versal integral invariants by Poincare and Cartan are elucidated. Examples of the 'universal invariant solutions' of the Schrodinger equation, i.e. se lf-consistent eigenstates of the universal integrals of motion, are given. Applications to the physics of optical and particle beams are discussed.