A technique is presented to scrutinize a piece of remnants of invariants [R. B. Shirts and W. P. Reinhardt, J. Chem. Phys. 77, 15 (1982)] buried in chaos in many degrees of freedom (DOF) dynamical systems in terms of canonical perturbation theory based on Lie transforms. The transformed canonical variables are often evaluated by the truncation of the coordinate transformation at a finite order in the original Hamiltonian system. However, the truncation of canonical variables gives rise to a loss of the symplectic property of the system. This results in apparent abrupt fluctuation of the action integrals, which yields a misinterpretation. We demonstrate, in a three-DOF Hamiltonian system of HCN isomerization reaction, that our technique can detect remnants of invariants buried in the potential well even at energies higher than the potential barrier, although the conventional truncation scheme fails to do so. This technique makes it possible to shed light on the physical insight into how the reactive mode exchanges its energy with the other modes and through which resonance the energy exchange takes place in reacting systems.