Two-center problem with harmonic-like interactions: periodic orbits and non-integrability

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 and Lyapunov exponents. Explicit periodic orbits bifurcating from the equilibrium points are presented as well.