We develop a Liouville perturbation theory for weakly driven and weakly open quantum systems in situations when the unperturbed system has a number of conservations laws. If the perturbation violates the conservation laws, it drives the system to a new steady state which can be approximately but efficiently described by a (generalized) Gibbs ensemble characterized by one Lagrange parameter for each conservation law. The value of those has to be determined from rate equations for conserved quantities. Remarkably, even weak perturbations can lead to large responses of conserved quantities. We present a perturbative expansion of the steady state density matrix; first we give the condition that fixes the zeroth order expression (Lagrange parameters) and then determine the higher order corrections via projections of the Liouvillian. The formalism can be applied to a wide range of problems including two-temperature models for electron-phonon systems, Bose condensates of excitons or photons or weakly perturbed integrable models. We test our formalism by studying interacting fermions coupled to non-thermal reservoirs, approximately described by a Boltzmann equation. In equilibrium many particle systems can be efficiently described by Gibbs ensembles, characterized by one Lagrange parameter (inverse temperature, chemical potential) for each exactly conserved quantity (energy, particle number). Such a simple but powerful description is, in general, absent for driven non-equilibrium systems. As long as weak perturbations, which drive the system out of equilibrium, have small effects one can resort to perturbation theory. Kubo formulas, for example, describe the response to time-dependent Hamiltonian perturbations. Also for open systems described by Lindblad operators similar formulations of perturbation theory can be developed [1-9]. If the density matrix of the unperturbed system is not unique (as is the case for all Hamiltonian systems) one has to use degenerate Liouville perturbation theory [1,7].There are many situations where even weak perturbations of interacting many-particle systems can have large effects. A famous example is the Bose-Einstein condensation of excitons and polaritons [10][11][12]: irradiation by light creates more and more excitons which equilibrate approximately and form a Bose-Einstein condensate. Importantly, the exciton number is approximately conserved thus even weak pumping can compensate for exciton losses and leads to large densities of these excitations. Similarly, condensations of photons or magnons have been observed [13][14][15][16][17]. This is a general phenomenon: whenever approximate conservation laws exist, small perturbations which weakly break those conservation laws can drive the system out of equilibrium, to a steady state completely different from the initial one. In the example given above, the approximately conserved quantity is the exciton number. Other examples use the approximate conservation of spin to induce large nuclear spin polarization [18] for medical applications o...