  The Lindhard-Winther (LW) equipartition sum rule shows that within the linear
response theory, the stopping power of an energetic point-charge projectile in
a degenerate electron gas medium, receives equal contributions from
single-particle and collective excitations in the medium. In this paper we show
that the LW sum rule does not necessarily hold for an extended projectile ion
and for ion-clusters moving in a fully degenerate electron gas. We have derived
a generalized equipartition sum rule and some related sum rules for this type
of projectiles. We also present numerical plots for He$^{+}$ ion and He$^{+}$
ion-clusters.
