  We discuss consequences of assuming ($i$) that the (Majorana) neutrino mass
matrix $M_\nu$ displays flavour antisymmetry, $S_\nu^T M_\nu S_\nu=-M_\nu$ with
respect to some discrete symmetry $S_\nu$ contained in $SU(3)$ and ($ii$)
$S_\nu$ together with a symmetry $T_l$ of the Hermitian combination
$M_lM_l^\dagger$ of the charged lepton mass matrix forms a finite discrete
subgroup $G_f$ of $SU(3)$ whose breaking generates these symmetries. Assumption
($i$) leads to at least one massless neutrino and allows only four textures for
the neutrino mass matrix in a basis with a diagonal $S_\nu$ if it is assumed
that the other two neutrinos are massive. Two of these textures contain a
degenerate pair of neutrinos.Assumption ($ii$) can be used to determine the
neutrino mixing patterns. We work out these patterns for two major group series
$\Delta(3 N^2)$ and $\Delta(6 N^2)$ as $G_f$. It is found that all $\Delta(6
N^2)$ and $\Delta(3 N^2)$ groups with even $N$ contain some elements which can
provide appropriate $S_\nu$. Mixing patterns can be determined analytically for
these groups and it is found that only one of the four allowed neutrino mass
textures is consistent with the observed values of the mixing angles
$\theta_{13}$ and $\theta_{23}$. This texture corresponds to one massless and a
degenerate pair of neutrinos which can provide the solar pair in the presence
of some perturbations. The well-known groups $A_4$ and $S_4$ provide examples
of the groups in respective series allowing correct $\theta_{13}$ and
$\theta_{23}$. An explicit example based on $A_4$ and displaying a massless and
two quasi degenerate neutrinos is discussed.
