Invariant manifolds of dynamical systems close to a rotation: Transverse to the rotation axis

We consider one parameter families of vector fields depending on a parameter ? such that for ?=0 the system becomes a rotation of R²×Rⁿ around {0}×Rⁿ and such that for ?>0 the origin is a hyperbolic singular point of saddle type with, say, attraction in the rotation plane and expansion in the complementary space. We look for a local subcenter invariant manifold extending the stable manifolds to ?=0. Afterwards the analogous case for maps is considered. In contrast with the previous case the arithmetic properties of the angle of rotation play an important role.