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In the considered models the flow pattern near current layer with attached MHD shock waves is not prescribed but is determined from a self-consistent solution of the MHD problem in the approximation of a strong magnetic field. Generalized analytical solutions are found taking into account the possibility of a current layer rupture in the region of anomalous plasma resistivity. The global structure of the magnetic field in the reconnection region and its local properties near the current layer and attached discontinuities are studied. In the reconnection regime with reverse current in the current layer, the attached discontinuities occur to be trans-Alfvenic shock waves near the current layer edges. Two types of transition of nonevolutionary shocks into evolutionary ones along discontinuous flows are shown to be possible, depending on the geometrical model parameters. The relationship between the results obtained and numerical magnetic reconnection experiments is discussed.