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The coupled nonlinear Schrödinger (CNLS) system is often used to describe physical problems such as wave propagation in a birefringent optical fiber. Under investigation in this paper is the nonlocal -symmetric reverse-space type of CNLS system, through constructing two types of Darboux transformation (DT), we will derive a new series of nonlocal analytic solutions: (1) Single-periodic and double-periodic solutions; (2) Four different combinations about dark and anti-dark solitons, as well as the bright and dark breathers on a periodic background. Moreover, the fundamental properties and dynamical behaviors of those solutions will be discussed. [ABSTRACT FROM AUTHOR]