  We discuss the pairing symmetry of a cobaltate superconductor
Na$_x$CoO$_2\cdot y$ H$_2$O by adopting an effective single band model that
takes into account the $e_g'$ hole pockets, as discussed in our previous paper
[to appear in Phys. Rev. Lett.] Here we consider the off-site repulsions in
addition to the on-site repulsion considered in our previous study. We show
that the spin-triplet f-wave pairing proposed in our previous study is robust
to some extent even in the presence of off-site repulsions. However, f-wave
pairing gives way to singlet pairings for sufficiently large values of off-site
repulsions. Among the singlet pairings, i-wave and extended s-wave pairings are
good candidates which do not break time reversal symmetry below $T_c$ in
agreement with the experiments.
