  The virialization of matter overdensities within dark energy subsystems is
considered under the restrictive assumptions (i) spherical-symmetric density
profiles, (ii) time-independent quintessence equation of state parameter, w,
and (iii) nothing but gravitational interaction between dark energy scalar
field and matter. In addition, the quintessence subsystem is conceived as made
of ``particles'' whose mutual interaction has intensity equal to G(1+3w) and
scales as the inverse square of their distance. Then the virial theorem is
formulated for subsystems. In the special case of fully clustered quintessence,
energy conservation is assumed with regard to either the whole system (global
energy conservation), or to the matter subsystem within the tidal potential
induced by the quintessence subsystem (partial energy conservation). Further
investigation is devoted to a few special values, w=-1/3, -1/2, -2/3, -1. The
special case of fully clustered (i.e. collapsing together with the matter)
quintessence is studied in detail. The general case of partially clustered
quintessence is considered in terms of a degree of quintessence de-clustering,
\zeta, ranging from fully clustered (\zeta=0) to completely de-clustered
(\zeta=1) quintessence, respectively. The special case of unclustered (i.e.
remaining homogeneous) quintessence is also discussed. The trend exhibited by
the fractional (virialization to turnaround) radius, \eta, as a function of
other parameters, is found to be different from its counterparts reported in
earlier attempts. The reasons of the above mentioned discrepancy are discussed.
