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 unex-pected 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 a priori non hamiltonian systems is very similar to the behavior of usual quasi-integrable systems. The equations. Euler’s problem in the plane (see Figure 1) is defined by the system of differential equations The two fixed centers are the points (1 , 0) and ( − 1 , 0), and the moving particle is the point ( x, y ). We have set x A x 1, x B The problem can be defined in the 3-dimensional space in the same way, and is also integrable, as was noticed by Euler. However, we will restrict ourselves to the planar case.