Nilpotent centers from analytical systems on center manifolds

Consider analytical three-dimensional differential systems having a singular point at the origin such that its linear part is y∂x−λz∂z for some λ≠0. The restriction of to Center Manifold has nilpotent origin. We prove if restricted system analytic and center origin, with Andreev number 2, then admits formal inverse Jacobi multiplier. also centers systems, on manifolds, are limits Hopf-type centers. use these results solve problem without restricting parametrization manifold.