  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 some
assumptions , we show its equivalence to a normal form under a local
biholomorphism at the singularity. 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 investigate the
existence of a complex submanifold of positive dimension in C n that intersects
a real submanifold along two totally and real analytic submanifolds that
intersect transversally at a possibly non-isolated CR singularity.
