The main properties of the electron Bernstein-Greene-Kruskal solitary waves (electron holes) are investigated prescribing the model potential shapes. The analytical expressions for the trapped particles distributions and the width dependence on the amplitude are obtained following the classic BGK method. It is shown that in a stationary kinetic consideration there is no limitation on the hole Mach number. Such a limitation arises only from the conditions of the BGK wave stability. The temporal evolution of the electron holes is simulated using the obtained trapped particles distributions and some qualitative conclusions about their stability are presented. © 1984 IOP publishing Ltd.