  The flavour neutrino puzzle is often addressed by considering neutrino mass
matrices $m$ with a certain number of vanishing entries ($m_{ij}=0$ for some
values of the indices), since a reduction in the number of free parameters
increases the predictive power. Symmetries that can enforce textures zero can
also enforce a more general type of conditions $f(m_{ij})=0$ with $f$ some
function of the matrix elements $m_{ij}$. In this case $m$ can have all entries
non-vanishing with no reduction in its predictive power. We classify all
generation-dependent $U(1)$ symmetries which, in the presence of two leptonic
Higgs doublets, can reduce the number of independent high-energy parameters of
type-I seesaw to the minimum number compatible with non-vanishing neutrino
mixings and CP violation. These symmetries are broken above the scale where the
effective operator is generated and can thus remain covert, in the sense that
no explicit evidence of the symmetry can be read off the neutrino mass matrix,
and different symmetries can give rise to the same low-energy structure. We
find that only two cases are viable: one yields a structure with two
zero-textures already considered in the literature, the other has no
zero-textures and has never been considered before. It predicts normal
ordering, a lightest neutrino mass $ \sim 10$ meV, a Dirac phase $\delta \sim
\frac{3\pi}{2}$ and definite values for the Majorana phases.
