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We consider a Majorana neutrino mass matrix $\mathcal{M}_\nu$ with$(\mathcal{M}_\nu)_{\mu\mu} = (\mathcal{M}_\nu)_{\tau\tau} = 0$, in the basiswhere the charged-lepton mass matrix is diagonal. We show that this pattern forthe lepton mass matrices can be enforced by extending the Standard Model withthree scalar SU(2) triplets and by using a horizontal symmetry group$\mathbbm{Z}_4$. The Ma--Sarkar (type-II seesaw) mechanism leads to very smallvacuum expectation values for the triplets, thus explaining the smallness ofthe neutrino masses; at the same time, that mechanism renders the physicalscalars originating in the triplets very heavy. We show that the conditions$(\mathcal{M}_\nu)_{\mu\mu} = (\mathcal{M}_\nu)_{\tau\tau} = 0$ allow both fora normal neutrino mass spectrum and for an inverted one. In the first case, theneutrino masses must be larger than $0.1 {eV}$ and the atmospheric mixing angle$\theta_{23}$ must be practically equal to $45^\circ$. In the second case, theproduct $\sin{\theta_{13}} | \tan{2 \theta_{23}} |$ must be of order one orlarger, thus correlating the large or maximal atmospheric neutrino mixing withthe smallness of the mixing angle $\theta_{13}$.Comment: 13 pages, no figures, plain LaTeX; one equation added, published  references updated, final version for J. Phys.

On a model with two zeros in the neutrino mass matrix