  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.
