More than 200 cases of energetic ion beams with an energy spectrum consisting of 1--3 narrow lines were observed during a period from August 1995 to August 1998 in the Earth's magnetosheath and in the region upstream of the Earth's bow shock. The observations are from the DOK-2 experiment on board of the Interball-1 spacecraft. Because the relative width at half maximum of these lines is of only 15--30%, we use the term Almost Monoenergetic Ions (AMI) for these events. Ion energy values varied for different events from 30 to 600 keV but were almost unchanged during each event. In two peak spectra the energy values ratio was 1:2 and in three peak spectra the ratios were 1:2:(5--6). Such line spectra cannot be explained by current models of particle acceleration or escape from the magnetosphere. We propose a hypothesis explaining the origin and main features of AMI as solar wind ions acceleration in a strong electrostatic field burst in a small region, possibly on the magnetopause or on the bow shock. ¿ 1999 American Geophysical Union