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It is well known that, for linear systems, the model matching problem is equivalent to a disturbance decoupling problem with disturbance measurement. The solution of both problems can be expressed in terms of properties of invariant subspaces of the system and the model. In this paper, it is shown that, for nonlinear systems, under appropriate hypotheses, analogous results can be obtained. The solvability of a model matching problem can be expressed in terms of properties of a suitable invariant distribution. It is shown that, as for linear systems, those properties are related to the "infinite zero structures" of the system and the model.