  We discuss an even-odd effect for an impurity with an $N$-fold degenerate
internal states immersed in a two-dimensional superfluid--Mott-insulator
quantum critical bath, which is described by an spin-$S$ XY Bose-Kondo impurity
model with $N=2S+1$. Using a dimensional- and momentum-cut off regularized
renormalization group and unbiased large-scale Monte Carlo numerical
simulations, we establish the phase diagram for the $S=1$ impurity with all the
relevant terms included. We show that the $S=1$ impurity with 3-fold degeneracy
is fully screened by the critical bath, in qualitative contrast to the spin-1/2
case where the impurity is only partially screened. We then argue that all
impurities with odd 2$S$ share the same universal physics as the spin-1/2 case,
and all impurities with even 2$S$ are as the spin-1 case. We validate our
conjecture with unbiased Monte Carlo simulations up to $S=2$. A physical
consequence of this even-odd effect is that two $N=2$ degenerate impurities in
the critical bath form a bound state at a sufficiently low temperature, which
can be realized using ultracold atoms in an optical lattice.
