Finite dimensional Hamiltonians and almost-periodic solutions for 2+1 dimensional three-wave equations

For the 2+1 dimensional three-wave equation, by using the known nonlinear constraints from 2+1 dimensions to 1+1 dimensions, we reduce it further to 0+1 dimensional (finite dimensional) Hamiltonian systems with constraints of Neumann type. These Hamiltonian systems are proved to be Lionville integrable by finding a full set of involutive conserved integrals and proving their functional independence. Moreover, almost-periodic solutions of the 2+1 dimensional three-wave equation are obtained by solving these Hamiltonian systems explicitly.