Abstract. 
In this work, we study a model of simplified four-body problem called planar
two-center-two-body problem. In the plane, we have two fixed centers
Q_1=(-\chi,0), Q_2=(0,0) of masses 1, and two moving bodies Q_3 and Q_4
of masses \mu\ll 1. They interact via Newtonian potential. Q_3 is captured by
Q_2, and Q_4 travels back and forth between two centers. Based on a model
of Gerver, we prove that there is a Cantor set of initial conditions which lead
to solutions of the Hamiltonian system whose velocities are accelerated to
infinity within finite time avoiding all early collisions. We consider
this model as a simplified model for the planar four-body problem case 
of the Painlevé conjecture. This is a joint work with Dmitry Dolgopyat.