Based on electron diffraction experimental observations the reconstructions of the indium-rich indium antimonide and indium arsenide (001) surfaces are assigned as c(8x2) or closely related c(8x2)/(4x2). At room temperature scanning tunneling and atomic force microscopy studies also evidence highly symmetric c(8x2) or (4x2) reconstructions. However, microscopic studies done at cryogenic temperatures indicate lowering of the surface structures symmetry from c2mm/p2mm to p2, structural domains, disorder and fluctuations. In the present paper we will show that the surfaces are well described by a so called zeta-family models with certain atomic rows occupied slightly above 50%. Atomic vacancies are confined to these rows and rapidly fluctuate at room temperature. Averaging effects cause that experiments done at elevated temperatures using slow methods evidence symmetric structures. In contrast, at low temperatures, the vacancies stabilize to form regular double-period structures along the rows, but this spontanoeusly leads to the complete surface structure having only p2 symmetry group, structural domains and partial disorder on the surface. We have also identified a variety of local structures appearing at the domain walls. This complex surface structure is inherent to the thermodynamic equilibrium of the system as indicated by failed attempts to increase surface indium content by adsorption of In atoms from a gas phase. Density functional theory calculations confirm that the new surface structure is a minimum energy configuration at fixed stoichiometry. Simulated scanning tunneling microscopy images confirm proposed model of the structure.