  We study a germ of real analytic $n$-dimensional submanifold of ${\mathbf
C}^n$ 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 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. If the
singularity is {\it abelian}, we show, under some assumptions on the linear
part of $\sigma$ at the singularity, that the real submanifold is
holomorphically equivalent to an analytic normal form. We also show that if a
real submanifold is formally equivalent to a quadric, it is actually
holomorphically equivalent to it, if a small divisors condition is satisfied.
Finally, we prove that, in general, there exists a complex submanifold of
positive dimension in ${\mathbf C}^n$ that intersects a real submanifold along
two totally and real analytic submanifolds that intersect transversally at a CR
singularity of the {\it complex type}.
