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The direction vector of the first rotating axis near the fixed platform remains constant in most existing 2R parallel mechanisms, while the direction vector of the second rotating axis near the moving platform only changes around the first rotating axis, and it remains constant relative to the moving platform. Based on spherical 4R mechanism, a class of 1Rv parallel mechanism with variable/invariable rotation axis was synthesized, and the change of the rotation axis of 1Rv mechanism was derived according to the finite screw theory. Based on bifurcation 1Rv parallel mechanisms with variable/invariable rotation axis, a 2Rv parallel mechanisms with two variable/invariable rotational axes was proposed by using the finite screw theory. The assembly condition and drive configuration of this kind of 2Rv parallel mechanism were analyzed. The novel bifurcated 2Rv parallel mechanism consists of four motion modes: invariable-invariable axes motion mode (in this mode, the first axis of rotation vector was invariable, and the second axis of rotation was also invariable relative to the moving platform), invariable-variable axes motion mode (in this mode, the first axis of rotation vector was invariable, and the second axis of rotation was variable relative to the moving platform), variable-invariable axes motion mode (in this mode, the first axis of rotation vector was variable, and the second axis of rotation was invariable relative to the moving platform), and variable-variable axes motion mode (in this mode, the first axis of rotation vector was variable, and the second axis of rotation was also variable relative to the moving platform). The existing 2R parallel mechanism with two invariable rotation axes was extended to the bifurcation 2Rv generalized parallel mechanism with two variable/invariable rotation axes (variable and invariable rotation axes).