The paper deals with perturbations of the equation that have a number of conservation laws. When a small term is added to the equation its conserved quantities usually decay at individual rates, a phenomenon known as a selective decay. These rates are described by the simple law using the conservation laws’ generating functions and the added term. Yet some perturbation may retain a specific quantity(s), such as energy, momentum and other physically important characteristics of solutions. We introduce a procedure for finding such perturbations and demonstrate it by examples including the KdV–Burgers equation and a system from magnetodynamics. Some interesting properties of solutions of such perturbed equations are revealed and discussed. DOI: 10.1134/