Reduction of bi-Hamiltonian systems and separation of variables: An example from the Boussinesq hierarchy

We discuss the Boussinesq system with the stationary time $t_5$ within a general framework of stationary flows of $n$-Gel'fand–Dickey hierarchies. A careful use of the bi-Hamiltonian structure can provide a set of separation coordinates for the corresponding Hamilton–Jacobi equations.