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−→Z→HA followed by invisible decays A→JJJ and H→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 bb¯bb¯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.