  Flavour symmetries appropriate for describing a neutrino spectrum with
degenerate solar pair and a third massive or massless neutrino are discussed.
We demand that the required residual symmetries of the leptonic mass matrices
be subgroups of some discrete symmetry group $G_f$. $G_f$ can be a subgroup of
SU(3) if the third neutrino is massive and we derive general results on the
mixing angle predictions for various discrete subgroups of SU(3). The main
results are: (a) All the SU(3) subgroups of type C fail in simultaneously
giving correct $\theta_{13}$ and $\theta_{23}$. (b) All the groups of type D
can predict a relation $\cos^2\theta_{13} \sin^2\theta_{23}=\frac{1}{3}$ among
the mixing angles which appears to be a good zeroth order approximation. Among
these, various $\Delta(6n^2)$ groups with $n\geq 8$ can simultaneously lead
also to $\sin^2 \theta_{13}$ in agreement with global fit at 3$\sigma$. (c) The
group $\Sigma(168)\cong PSL(2,7)$ predicts near to the best fit value for
$\theta_{13}$ and $\theta_{23}$ within the 1$\sigma$ range. All discrete
subgroups of U(3) with order $<512$ and having three dimensional irreducible
representation are considered as possible $G_f$ when the third neutrino is
massless. Only seven of them are shown to be viable and three of these can
correctly predict $\theta_{13}$ and/or $\theta_{23}$. The solar angle remains
undetermined at the leading order in all the cases due to degeneracy in the
masses. A class of general perturbations which can correctly reproduce all the
observables are discussed in the context of several groups which offer good
leading order predictions.
