Equilibrium solutions of an electron beam injected into a plasma are considered. This is a preliminary step in studying the stability of the so‐called ``E layer'' of the Astron thermonuclear reactor. The model consists of a plane sheet of streaming electrons in a homogeneous plasma. Equilibrium solutions are obtained from the collisionless Boltzmann equations for three fluids: one of stream electrons, another of plasma electrons, and a third of plasma ions. Two important cases are treated in this manner: cold plasma and cold beam. The distribution of beam electrons, plasma ions, and plasma electrons are disclosed.