Lepton number violation decays l(i) --> l(j)gammagamma and l(j) --> 3l(j) are investigated in the framework of E6 superstring model and the explicit expression for these decay rates are obtained. Relations between lepton number violation decays l(i) --> l(j)gammagamma and l(i) --> l(j)gamma are established and it is shown that the relative value of these decays is independent of the details of the considered model. In the considered model we predict that BR(tau --> mugammagamma) is large than BR(mu --> egamma) in spite of the fact that tau --> mugammagamma decay is suppressed by a factor of alpha(el)/pi, compared to the mu --> --> egamma. Using the present experimental restrictions, the upper bound for tau --> mugammagamma decays and lower limits for the masses of new particles are obtained. Also it is shown that rare mu --> 3e decay has large BR in comparison with tau --> 3mu and tau --> 3e decays.