For an integer linear optimization (ILO) problem, persistency of its linear optimization (LO) relaxation is a property that for every optimal solution of the relaxation that assigns integer values to some variables, there exists an optimal solution of the ILO problem in which these variables retain the same values. Although persistency has been used to develop heuristic, approximation, and fixed-parameter algorithms for special cases of ILO, its applicability remains unknown in the literature. In this paper we reveal a maximal subclass of ILO such that its LO relaxation has persistency. Specifically, we show that the LO relaxation of ILO on unit-two-variable-per-inequality (UTVPI) systems has persistency and is (in a certain sense) maximal among such ILO. Our persistency result generalizes the results of Nemhauser and Trotter, Hochbaum et al., and Fiorini et al. Even more, we propose a stronger property called \emph{neighborhood persistency} and show that the LO relaxation of ILO on UTVPI systems in general has this property. Using this stronger result, we obtain a fixed-parameter algorithm (where the parameter is the solution size) and another proof of two-approximability for ILO on UTVPI systems where objective functions and variables are non-negative.