  In the equilibrium statistical mechanics of classical Coulomb fluids, the
long-range tail of the Coulomb potential gives rise to the Stillinger-Lovett
sum rules for the charge correlation functions. For the jellium model of mobile
particles of charge $q$ immersed in a neutralizing background, the fixing of
one of the $q$-charges induces a screening cloud of the charge density whose
zeroth and second moments are determined just by the Stillinger-Lovett sum
rules. In this paper, we generalize these sum rules to the screening cloud
induced around a pointlike guest charge $Z q$ immersed in the bulk interior of
the 2D jellium with the coupling constant $\Gamma=\beta q^2$ ($\beta$ is the
inverse temperature), in the whole region of the thermodynamic stability of the
guest charge $Z>-2/\Gamma$. The derivation is based on a mapping technique of
the 2D jellium at the coupling $\Gamma$ = (even positive integer) onto a
discrete 1D anticommuting-field theory; we assume that the final results remain
valid for all real values of $\Gamma$ corresponding to the fluid regime. The
generalized sum rules reproduce for arbitrary coupling $\Gamma$ the standard
Z=1 and the trivial Z=0 results. They are also checked in the Debye-H\"uckel
limit $\Gamma\to 0$ and at the free-fermion point $\Gamma=2$. The generalized
second-moment sum rule provides some exact information about possible sign
oscillations of the induced charge density in space.
