Birational properties of tangent to the identity germs without non-degenerate singular directions

We provide a family of isolated tangent to the identity germs (Formula presented.) which possess only degenerate characteristic directions, and for which the lift of (Formula presented.) to any modification (with suitable properties) has only degenerate characteristic directions. This is in sharp contrast with the situation in dimension 2, where any isolated tangent to the identity germ (Formula presented.) admits a modification where the lift of (Formula presented.) has a non-degenerate characteristic direction. We compare this situation with the resolution of singularities of the infinitesimal generator of (Formula presented.), showing that this phenomenon is not related to the non-existence of complex separatrices for vector fields of Gomez-Mont and Luengo. Finally, we describe the set of formal (Formula presented.) -invariant curves, and the associated parabolic manifolds, using the techniques recently developed by López-Hernanz et al.