  Using a data sample of 980 fb$^{-1}$ collected with the Belle detector at the
KEKB asymmetric-energy $e^+e^-$ collider, we study the processes of $\Xi^0_c\to
\Lambda\bar K^{*0}$, $\Xi^0_c\to \Sigma^0\bar K^{*0}$, and $\Xi^0_c\to
\Sigma^+K^{*-}$ for the first time. The relative branching ratios to the
normalization mode of $\Xi^0_c\to\Xi^-\pi^+$ are measured to be $${\cal
B}(\Xi^0_c\to \Lambda\bar K^{*0})/{\cal B}(\xic\to \Xi^-\pi^+)=0.18\pm0.02({\rm
stat.})\pm0.01({\rm syst.}),$$ $${\cal B}(\Xi^0_c\to \Sigma^0\bar K^{*0})/{\cal
B}(\xic\to \Xi^-\pi^+)=0.69\pm0.03({\rm stat.})\pm0.03({\rm syst.}),$$ $${\cal
B}(\Xi^0_c\to \Sigma^+K^{*-})/{\cal B}(\xic\to \Xi^-\pi^+)=0.34\pm0.06({\rm
stat.})\pm0.02({\rm syst.}),$$ where the uncertainties are statistical and
systematic, respectively. We obtain %measure the branching fractions of
$\Xi^0_c\to \Lambda\bar K^{*0}$, $\Xi^0_c\to \Sigma^0\bar K^{*0}$, and
$\Xi^0_c\to \Sigma^+K^{*-}$ to be $${\cal B}(\Xi^0_c\to \Lambda\bar
K^{*0})=(3.3\pm0.3({\rm stat.})\pm0.2({\rm syst.})\pm1.0({\rm
ref.}))\times10^{-3},$$ $${\cal B}(\Xi^0_c\to \Sigma^0\bar
K^{*0})=(12.4\pm0.5({\rm stat.})\pm0.5({\rm syst.})\pm3.6({\rm
ref.}))\times10^{-3},$$ $${\cal B}(\Xi^0_c\to \Sigma^+K^{*-})=(6.1\pm1.0({\rm
stat.})\pm0.4({\rm syst.})\pm1.8({\rm ref.}))\times10^{-3},$$ where the
uncertainties are statistical, systematic, and from ${\cal B}(\xic \to
\Xi^-\pi^+)$, respectively. The asymmetry parameters $\alpha(\Xi^0_c\to
\Lambda\bar K^{*0})$ and $\alpha(\Xi^0_c\to \Sigma^+K^{*-})$ are
$0.15\pm0.22({\rm stat.})\pm0.04({\rm syst.})$ and $-0.52\pm0.30({\rm
stat.})\pm0.02({\rm syst.})$, respectively, where the uncertainties are
statistical followed by systematic.
