  An analytic solution describing an ion-acoustic collisionless shock,
self-consistently with the evolution of shock-reflected ions, is obtained. The
solution extends the classic soliton solution beyond a critical Mach number,
where the soliton ceases to exist because of the upstream ion reflection. The
reflection transforms the soliton into a shock with a trailing wave and a foot
populated by the reflected ions. The solution relates parameters of the entire
shock structure, such as the maximum and minimum of the potential in the
trailing wave, the height of the foot, as well as the shock Mach number, to the
number of reflected ions. This relation is resolvable for any given
distribution of the upstream ions. In this paper, we have resolved it for a
simple "box" distribution. Two separate models of electron interaction with the
shock are considered. The first model corresponds to the standard Boltzmannian
electron distribution in which case the critical shock Mach number only
insignificantly increases from M=1.6 (no ion reflection) to M=1.8 (substantial
reflection). The second model corresponds to adiabatically trapped electrons.
They produce a stronger increase, from M=3.1 to M=4.5. The shock foot that is
supported by the reflected ions also accelerates them somewhat further. A
self-similar foot expansion into the upstream medium is also described
analytically.
