A new proof of Poincar\'e's result on the restricted three-body problem

The problem of nonintegrability of the circular restricted three-body problem is very classical and important in dynamical systems. In the first volume of his masterpieces, Henri Poincar\'e showed the nonexistence of a real-analytic first integral which is functionally independent of the Hamiltonian and real-analytic in a small parameter representing the mass ratio as well as in the state variables, in both the planar and spatial cases. However, his proof was very complicated and unclear. In this paper, we give a new and simple proof of a very similar result for both the planar and spatial cases, using an approach which the author developed recently for nearly integrable systems.