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Generalized linear variational inequality (GLVI) is an extension of the canonical linear variational inequality. In recent years, a recurrent neural network (NN) called general projection neural network (GPNN) was developed for solving GLVIs with simple bound (often box-type or sphere-type) constraints. The aim of this paper is twofold. First, some further stability results of the GPNN are presented. Second, the GPNN is extended for solving GLVIs with general linear equality and inequality constraints. A new design methodology for the GPNN is then proposed. Furthermore, in view of different types of constraints, approaches for reducing the number of neurons of the GPNN are discussed, which results in two specific GPNNs. Moreover, some distinct properties of the resulting GPNNs are also explored based on their particular structures. Numerical simulation results are provided to validate the results.