The pulsed injector of the low energy positrons for positron accumulator LEPTA has been constructed and tested at JINR. The injector is based on 22Na radioactive source. Positrons from the source are moderated in the solid neon and injected into positron trap, where they are accumulated during about 80 seconds. For injection the positrons are extracted by the pulsed electric field and accelerated up to the required energy. The injector will generate positrons of the energy of up to 10 keV at relative energy spread of 2•10-3, intensity of 108 - 109 particles per pulse and at injection pulse duration of 300 nsec. The cryogenic source of slow positrons has been tested with a test isotope 22Na of the initial activity of 80 MBk. The continuous slow positron beam with average energy spread of 1.2 eV, width of a spectrum 1 eV has been obtained. The achieved moderator efficiency is about 1%. The accumulation process in the positron trap was investigated with electron flux. The life time of the electrons in the trap, τlife ≥ 80 s and capture efficiency ε ~ 0.4 have been obtained. The maximum number of the accumulated particle was Nexper = 2*108 at the initial flux of 5•106 electrons per sec. The dynamics of slow positrons (electrons) in the injector and physics of the particle storage process are discussed in the report. The work is supported by RFBR grant No. 05-02-16320, the president of Russia Federation grant (MK-3948.2007.2) for supporting of young scientists and leading sceintific schools.