We explore the in-medium properties of heavy-quarkonium states at finite baryo-chemical potential and finite transverse momentum based on a modern complex valued potential model. Our starting point is a novel, rigorous derivation of the generalized Gauss law for in-medium quarkonium, combining the non-perturbative physics of the vacuum bound state with a weak coupling description of the medium degrees of freedom. Its relation to previous models in the literature is discussed. We show that our approach is able to reproduce the complex lattice QCD heavy quark potential even in the non-perturbative regime, using a single temperature dependent parameter, the Debye mass m D . After vetting the Gauss law potential with state-of-the-art lattice QCD data, we extend it to the regime of finite baryon density and finite velocity, currently inaccessible to first principles simulations. In-medium spectral functions computed from the Gauss law potential are subsequently used to estimate the ψ /J/ψ ratio in heavy-ion collisions at different beam energies and transverse momenta. We find qualitative agreement with the predictions from the statistical model of hadronization for the √ s N N dependence and a mild dependence on the transverse momentum.