  The problem of nonintegrability of the circular restricted three-body problem
is very classical and important in the theory of dynamical systems. It was
partially solved by Poincare in the nineteenth century: He showed that there
exists no real-analytic first integral which depends analytically on the mass
ratio of the second body to the total and is functionally independent of the
Hamiltonian. When the mass of the second body becomes zero, the restricted
three-body problem reduces to the two-body Kepler problem. We prove the
nonintegrability of the restricted three-body problem both in the planar and
spatial cases for any nonzero mass of the second body. Our basic tool of the
proofs is a technique developed here for determining whether perturbations of
integrable systems which may be non-Hamiltonian are not meromorphically
integrable near resonant periodic orbits such that the first integrals and
commutative vector fields also depend meromorphically on the perturbation
parameter. The technique is based on generalized versions due to Ayoul and Zung
of the Morales-Ramis and Morales-Ramis-Simo theories.
