  We present a search for new phenomena in events with two reconstructed $Z$
bosons and large missing transverse momentum, sensitive to processes
$p\bar{p}\rightarrow X_2X_2 \rightarrow Z Z X_1 X_1$, where $X_2$ is an
unstable particle decaying as $X_2\rightarrow ZX_1$ and $X_1$ is undetected.
The particles $X_1$ and $X_2$ may be, among other possibilities, fourth
generation neutrinos or supersymmetric particles. We study the final state in
which one $Z$ boson decays to two charged leptons and the second decays
hadronically. In data corresponding to an integrated luminosity of 4.2
fb$^{-1}$ from proton-antiproton collisions recorded by the CDF II detector at
the Tevatron, with center-of-mass energy of 1.96 TeV, we find agreement between
data and standard-model backgrounds. We calculate 95% confidence level upper
limits on the cross section of the process $p\bar{p}\rightarrow X_2X_2
\rightarrow Z Z X_1 X_1$ ranging from 50 fb to 1 pb, depending on the masses of
$X_1$ and $X_2$.
