On solutions of simplified systems of nonlinear differential equations related to the problem of four bodies

In the main part we consider a system describing the motion of four bodies under the action of gravity. Using the set of elementary algebraic transformations, a simplified system consisting of nonlinear differential equations, each of which has the second order was developed. 21 autonomous nonlinear differential equation of the first order with respect to one of the components of the system, whose general solution has the Painlevé property were obtained. Necessary and sufficient conditions for Painlevé properties in the studied system were established.