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The nonlinear equivalences of both finite and infinite zero structures of linear systems have been well understood for single input single output systems and have found many applications in nonlinear control theory. The extensions of these notions to multiple input multiple output systems have proven to be highly sophisticated. In this paper, we propose constructive algorithms for decomposing a nonlinear system that is affine in control. These algorithms require modest assumptions on the system and apply to general multiple input multiple output systems that do not necessarily have the same number of inputs and outputs. They lead to various normal form representations and reveal the structure at infinity, the zero dynamics and the invertibility properties, all of which represent nonlinear equivalences of relevant linear system structural properties.