  We discuss a neutrino mass matrix $\mathcal{M}_\nu$ originally found by Babu,
Ma, and Valle (BMV) and show that this mass matrix can be characterized by a
simple algebraic relation. From this relation it follows that atmospheric
neutrino mixing is exactly maximal while at the same time an arbitrary mixing
angle $\theta_{13}$ of the lepton mixing matrix $U$ is allowed and--in the
usual phase convention--CP violation in mixing is maximal; moreover, neither
the neutrino mass spectrum nor the solar mixing angle are restricted. We put
forward a simple seesaw extension of the Standard Model where the family lepton
numbers are softly broken by the Majorana mass terms of the right-handed
neutrino singlets and the BMV mass matrix results from a non-standard $CP$
symmetry.
