On the non-integrability of a fifth order equation with integrable two-body dynamics

We consider the fifth order partial differential equation (PDE) $u_{4x,t}-5u_{xxt}+4u_t+uu_{5x}+2u_xu_{4x}-5uu_{3x}-10u_xu_{xx}+12uu_x=0$, which is a generalization of the integrable Camassa-Holm equation. The fifth order PDE has exact solutions in terms of an arbitrary number of superposed pulsons, with geodesic Hamiltonian dynamics that is known to be integrable in the two-body case N=2. Numerical simulations show that the pulsons are stable, dominate the initial value problem and scatter elastically. These characteristics are reminiscent of solitons in integrable systems. However, after demonstrating the non-existence of a suitable Lagrangian or bi-Hamiltonian structure, and obtaining negative results from Painlev\'{e} analysis and the Wahlquist-Estabrook method, we assert that the fifth order PDE is not integrable.