Exact Results in Quantum Theory

We review application of integrability to non-equilibrium problems showing how the non-equilibrium dynamics of local observables is fixed by all the irreducible representations of the underlying symmetry of the model, that eventually produce quasi-local conserved operators. We show how to include such operators in a reduced density matrix and how they also allow for an exact description of the spin conductivity in spin chains and how this is usually a non-continuos function of the interaction strength.