  We study the classical planar two-center problem of a particle $m$ subjected
to harmonic-like interactions with two fixed centers. For convenient values of
the dimensionless parameter of this problem we use the averaging theory for
showing analytically the existence of periodic orbits bifurcating from two of
the three equilibrium points of the Hamiltonian system modeling this problem.
Moreover, it is shown that the system is generically non-integrable in the
sense of Liouville-Arnold. The analytical results are complemented by numerical
computations of the Poincar\'e sections as well as providing some explicit
periodic orbits.
