First Approximations of a Reversible ODE System of

Near its equilibrium point we consider a 4-dimentional reversible system of odinary differential equation, having small parameters, and such that 0 is a quadruple eigen-value of the linear part of the nonperturbed system with 4x4 Jordan block. This system appeared from Hydrodynamics after a reduction on the center manifold of the water-wade problem. We construct the Newton polyhedron of the system and using it we had found all truncated systems and sets where each of them is the first approximations. Five truncated systems was obtained. Four of them are explicitly integrable. The fifth (basic) truncated system is not integrable. We make its preliminary analysis.