In this paper a study of the most general form of cylindrical symmetric non-static space-times is given using direct integration and algebraic techniques. In this study we deal with the cases in which all eigenvalues of the Riemann tensor are different. From this study we show that when the cylindrical symmetric non-static space-times admit proper projective symmetry, it turns out to be a special class of the space-like flat Friedmann-Robertson-Walker model.
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