Abstract: Firstly, by  using the elimination theory in algebraic geometry, we  gave sufficient conditions for the existence of  invariant algebraic surfaces in a three\|wave interaction model. Secondly, we constructed an infinite number of HamiltonianPoisson 
structures of the system, the system was bi\|Hamiltonian. Finally, we used the Poincaré compactification technique in R³ to describe the dynamic behavior at infinity of the system.

Key words: invariant algebraic surface, biHamiltonian structure, Poincaré compactification