The analysis of EXAFS data aimed at extracting information on the local atomic structure of materials is discussed. To identify the structure of interest, one has to solve a system of Fredholm integral equations of the first kind. The situation is considered in which three functions, that characterize the structure under study, are to be determined from one equation. In practice, in this case a nonuniqueness arises in the reconstruction of the unknown functions within a prescribed accuracy, so that additional information has to be used. For a broad range of materials, the functions that are reconstructed possess a rather specific form; namely, these functions appear to be a sum of several narrow peaks that can be approximated by δ functions. This qualitative information about the functions is non-classical. The proposed new analysis procedure is based on the classical theory of ill-posed problems and on using the available non-classical information. The efficiency of the algorithms proposed is illustrated by applying them to model and experimental data of a test compound consisting of uranium, silicon and oxygen, whose structure was previously identified by other authors.