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We discuss consequences of assuming ($i$) that the (Majorana) neutrino massmatrix $M_\nu$ displays flavour antisymmetry, $S_\nu^T M_\nu S_\nu=-M_\nu$ withrespect 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 discretesubgroup $G_f$ of $SU(3)$ whose breaking generates these symmetries. Assumption($i$) leads to at least one massless neutrino and allows only four textures forthe neutrino mass matrix in a basis with a diagonal $S_\nu$ if it is assumedthat the other two neutrinos are massive. Two of these textures contain adegenerate pair of neutrinos.Assumption ($ii$) can be used to determine theneutrino 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 $\Delta(6N^2)$ and $\Delta(3 N^2)$ groups with even $N$ contain some elements which canprovide appropriate $S_\nu$. Mixing patterns can be determined analytically forthese groups and it is found that only one of the four allowed neutrino masstextures is consistent with the observed values of the mixing angles$\theta_{13}$ and $\theta_{23}$. This texture corresponds to one massless and adegenerate pair of neutrinos which can provide the solar pair in the presenceof some perturbations. The well-known groups $A_4$ and $S_4$ provide examplesof the groups in respective series allowing correct $\theta_{13}$ and$\theta_{23}$. An explicit example based on $A_4$ and displaying a massless andtwo quasi degenerate neutrinos is discussed.

Neutrino masses and mixing from flavour antisymmetry