Finite-dimensional integrable systems associated with the Davey-Stewartson I equation

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.