  We study a germ of real analytic n-dimensional submanifold of $C^n$ that has
a complex tangent space of maximal dimension at a CR singularity. Under the
condition that its complexification admits the maximum number of deck
transformations, we first classify holomorphically its quadratic CR
singularity. We then study its transformation to a normal form under the action
of local (possibly formal) biholomorphisms at the singularity. We first
conjugate formally its associated reversible map $\sigma$ to suitable normal
forms and show that all these normal forms can be divergent. We then construct
a unique formal normal form under a non degeneracy condition.
