  We perform a careful study of the neutral scalar sector of a model which
includes a singlet, a doublet, and a triplet scalar field under $SU(2)$. This
model is motivated by neutrino physics, since it is simply the most general
version of the seesaw model of neutrino mass generation through spontaneous
violation of lepton number. The neutral Higgs sector contains three CP-even and
one massive CP-odd Higgs boson $A$, in addition to the massless CP-odd majoron
$J$. The weakly interacting majoron remains massless if the breaking of lepton
number symmetry is purely spontaneous. We show that the massive CP-odd Higgs
boson may invisibly decay to three majorons, as well as to a CP-even Higgs $H$
boson plus a majoron. We consider the associated Higgs production $e^+e^- \to Z
\to H A$ followed by invisible decays $A \to JJJ$ and $H \to JJ$ and derive the
corresponding limits on masses and coupling that follow from LEP I precision
measurements of the invisible Z width. We also study a novel $b \bar{b}
b\bar{b}p\!\!\!/_T$ signal predicted by the model, analyse the background and
perform a Monte-Carlo simulation of the signal in order to illustrate the
limits on Higgs boson mass, couplings and branching ratios that follow from
that.
