  We present a supersymmetric model in which the lepton mixing matrix $U$
obeys, at the seesaw scale, the Harrison--Perkins--Scott
\textit{Ansatz}--vanishing $U_{e3}$, maximal atmospheric neutrino mixing, and
$\sin^2{\theta_\odot} = 1/3$ ($\theta_\odot$ is the solar mixing angle). The
model features a permutation symmetry $S_3$ among the three lepton multiplets
of each type--left-handed doublets, right-handed charged leptons, and
right-handed neutrinos--and among three Higgs doublets and three
zero-hypercharge scalar singlets; a fourth right-handed neutrino, a fourth
Higgs doublet, and a fourth scalar singlet are invariant under $S_3$. In
addition, the model has seven $\mathbbm{Z}_2$ symmetries, out of which six do
not commute with $S_3$. Supersymmetry is needed in order to eliminate some
quartic terms from the scalar potential, quartic terms which would make
impossible to obtain the required vacuum expectation values of the three Higgs
doublets and three scalar singlets. The Yukawa couplings to the charged leptons
are flavour diagonal, so that flavour-changing neutral Yukawa interactions only
arise at loop level.
