For the Davey-Stewartson I equation, which is an integrable equation in 1 + 2 dimensions, we have already found its Lax pair in (1 + 1)-dimensional form by nonlinear constraints. This paper deals with the second nonlinearization of this (1 + 1)-dimensional system to obtain three (1 + 0)-dimensional Hamiltonian systems with a constraint of Neumann type. The full set of involutive conserved integrals is obtained and their functional independence is proved. Therefore, the Hamiltonian systems are completely integrable in the Liouville sense. A periodic solution of the Davey-Stewartson I equation is obtained by solving these classical Hamiltonian systems as an example.
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