The center problem on a center manifold in R3R3

Fix a collection of polynomial vector fields on R3R3 with a singularity at the origin, for every one of which the linear part at the origin has two pure imaginary and one non-zero eigenvalue. We show that for each fixed value of the non-zero real eigenvalue the set of such systems having a center on the local center manifold at the origin corresponds to a variety in the space of admissible coefficients. We explicitly compute it for several families of systems with quadratic higher order terms.