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there are only a few words in this arbitrary piece of text from newspaper and journals. for better results and testing of the software use text from technical blogs. compatibility of systems of linear constraints over the set of natural numbers. criteria of compatibility of a system of linear and strict equations are considered. upper bounds for components of a minimal set of solutions. algorithms of construction of minimal generating sets of solutions for all types of systems are given. these criteria and the corresponding algorithms can be used in solving systems of all the considered types.