  Integrable systems offer rare examples of solvable many-body problems in the
quantum world. Due to the fine-tuned structure, their realization in nature and
experiment is never completely accurate, therefore effects of integrability are
observed only transiently. One way to overcome this limitation is to weakly
couple nearly integrable systems to baths and driving: these will stabilize
integrable effects up to arbitrary time and encode them in the stationary state
approximated by a generalized Gibbs ensemble. However, the description of such
driven dissipative nearly integrable models is challenging and no exact
analytical methods have been proposed so far. Here, we develop an iterative
scheme in which integrability breaking perturbations (baths) determine the
conserved quantities that play the leading role in a highly efficient truncated
generalized Gibbs ensemble description. Our scheme paves the way for easier
calculations in thermodynamically large systems and can be used to construct
unknown conserved quantities.
