  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 $\psi'/J/\psi$ 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 $\sqrt{s_{NN}}$
dependence and a mild dependence on the transverse momentum.
