Complex dynamical invariant for a PT- symmetric Hamiltonian system in higher dimensions

With a view to get further insight into the integrability of a dynamical system, we investigate the complex invariant for a three-dimensional Hamiltonian system using the extended complex phase space approach (ECPSA) characterized by x=x1+ip4,y=x2+ip5,z=x3+ip6,px=p1+ix4,py=p2+ix5 and pz=p3+ix6. For this purpose the rationalization method is utilized and the invariant obtained is expected to play an important role in the study of the complex trajectories for the system of concern.