Quasi-integrability in a class of systems generalizing the problem of two fixed centers

The problem of two fixed centers is a classical integrable problem, stated and integrated by Euler in 1760. The integrability is due to the unexpected first integral $G$. Some straightforward generalizations of the problem still have the generalization of $G$ as a first integral, but do not possess the energy integral. We present some numerical integrations suggesting that in the domain of bounded orbits the behavior of these {\it a priori} non hamiltonian systems is very similar to the behavior of usual quasi-integrable systems.