Existence of orbits homoclinic to an elliptic equilibrium, for a reversible system

We consider a reversible vector field in R-4, where the origin is a critical point, and where the differential at the origin has a pair of double non semisimple pure imaginary eigenvalues +/-i omega. We assume that the coefficient epsilon of a cubic term of the normal form is positive and close to 0, and that a certain coefficient of order 5 is negative. Then we show that there exist two reversible orbits homoclinic to the origin, of size root epsilon and such that they oscillate with a damping in 1/t when t tends towards +/-infinity. For obtaining such a result, we give explicitly the inverse of the linearized operator around the reversible homoclinics of the normal form, and solve the problem by a fired point argument.