We study germs of real analytic $n$-dimensional submanifold of $\mathbf{C}^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions, we first classify holomorphically the quadrics having this property. We then study higher order perturbations of these quadrics and their transformations to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We are led to study formal Poincaré–Dulac normal forms (non-unique) of reversible biholomorphisms. We exhibit a reversible map of which the normal forms are all divergent at the singularity. We then construct a unique formal normal form of the submanifolds under a non degeneracy condition.

中文翻译：

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CR奇异点II处最大复切线空间的实子流形

我们研究$ \ mathbf {C} ^ n $的实际解析$ n $维子流形的细菌，该子流形具有在CR奇点处最大维的复杂切线空间。在某些假设下，我们首先对具有此属性的二次曲面进行全同分类。然后，我们研究这些二次曲面的高阶扰动及其在局部（可能是形式）双全纯性在奇异性作用下的转换为正态形式。我们被引导研究可逆双同态的正式庞加莱-杜拉克正规形式（非唯一）。我们展示了一个可逆图，其正常形式在奇异点上都是发散的。然后，我们在非简并条件下构造子流形的独特形式正规形式。