  We investigate the discovery potential of cubic kilometer neutrino
observatories such as IceCube to set stringent limits on the forbidden decays
$\eta\to\nu_e\bar\nu_e$ and $\eta\to\nu_{\tau}\bar\nu_{\tau}$. The signatures
for these decays are cascade events resulting from the charged-current
reactions of $\nu_e, \nu_{\tau}$, $\bar\nu_e$ and $\bar\nu_{\tau}$ on nuclei in
such detectors. Background cascade events are mainly due to $\nu_e$'s from
atmospheric $\mu$, $K^+$, and $K^{0}_S$ decays and to a lesser extent from
atmospheric $\nu_{\mu}$ neutral current interactions with nuclei. A direct
upper limit for the branching ratio $\eta \to \nu_{e,\tau} \bar\nu_{e,\tau}$ of
$4.5\times10^{-4}$ at 90% CL can be achieved.
