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We show that the germ of a contact transformation can be written in a unique way as a product , where only depends on the derivative of and the derivative of is trivial. We show that each contact transformation with trivial derivative can be constructed solving a convenient Cauchy problem.
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Ana Rita Martins, Orlando Neto, On the group of germs of contact transformations. Port. Math. 72 (2015), no. 4, pp. 393–406DOI 10.4171/PM/1972