A search for rare lepton decays of the eta meson was performed using the WASA detector at CELSIUS. Two candidates for double Dalitz decay eta->e+e-e+e- events are reported with a background of 1.3+/-0.2 events. This allows to set an upper limit to the branching ratio of 9.7E-5 (90% CL). The branching ratio for the decay eta->e+e-gamma is determined to (7.8+/-0.5 stat+/-0.8 syst)E-3. An upper limit (90% CL) for the branching ratio for the eta->e+e- decay is 2.7E-5 and a limit for the sum of the eta->mu+mu-mu+mu- and eta->pi+pi-mu+mu- decays is 3.6E-4.
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