CP conserving solutions are apart, we disregard [*ab initio*]{} the question of non-trivial phase vacua.

Upper bounds have been derived for the mass of the lightest Higgs scalar in the MSSM that take into account the large one loop corrections [@Haber] as well as some two-loop effects [@ceqw]. This prediction is of paramount importance in view of the experimental searches of the Higgs scalar in the LEP 200 and the LHC. The (M+1)SSM has an additional contribution $\propto \left| \lambda \right| ^{2}$ to the Higgs (mass$)^2.$ It has been shown [@Ellis; @King; @Bine] that requiring the theory to remain perturbative up to the GUT scale reduces this additional contribution to $<O$(30 GeV). The upper bounds on the lightest Higgs boson mass are considerably lowered if one also imposes all phenomenological constraints needed for the model to be realistic. This has been done in the case of universal supergravity couplings [@ERS1; @King] and also for some special non-universal models [@Brax]. Note that a relatively small increase in the Higgs mass may affect the experimental signatures. Since the predictions of the phenomenologically constrained (M+1)SSM for the Higgs sector has been the subject of a recent publication [@ERS2], we shall only give a 