The work aims to accurately solve the nonlinear free vibration response of a packaging system based on Lie integral method. Considering the nonlinear relation of the cubic polynomial form in the spring restoring force, the system model with single degree of freedom and damping dynamics was established based on the analytical mechanics principle. The differential equation Lie group transformation was used for the first time to solve the symmetry and two first integrals of the system, for the purpose of proving that, the exact solution of the free vibration of the packaging system was a class of elliptic integral functions when structural design parameters satisfied certain relations. The actual numerical simulation calculations showed that, the self-resonance frequency of the system increased with the increase of the initial amplitude conditions, and the nonlinear coefficient made the attenuation of the displacement amplitude faster. From the perspective of deduction process, the Lie symmetry theory is applied to the study on the nonlinear dynamic characteristics of packaging system, and the nonlinear coefficient and damping coefficient of the system do not need to satisfy the hypothesis of small parameters; therefore, the application scope is wider.