  We study the mass shifts of the charmonium ($\bar{c}c$) states ($J/\psi$,
$\psi(2S)$, $\psi(1D)$, $\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$) as well as
the bottomonium ($\bar{b}b$) states ($\Upsilon(1S)$, $\Upsilon(2S)$,
$\Upsilon_2(1D)$, $\chi_{b0}$, $\chi_{b1}$ and $\chi_{b2}$) in isospin
asymmetric nuclear matter. These are investigated using a generalized linear
sigma model. The broken scale invariance of QCD is incorporated in the chiral
$SU(2)\times SU(2)$ Lagrangian through an effective potential involving
logarithmic terms of a scalar (glueball) dilaton field $\chi$. The mass shifts
of the quarkonium states are obtained through the medium modifications of the
dilaton field which simulates the scalar gluon condensate of QCD. We observe an
appreciable mass drop in the states of heavy quarkonia under this study. The
in-medium masses at finite densities thus obtained should modify the in-medium
partial decay widths of heavy quarkonia to open heavy flavor mesons. These
density effects can be probed in in the high energy nuclear collisions at the
future facility at GSI (at Germany) and JINR (at Russia) in the experiments
producing highly dense baryonic matter.
