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A CR manifold $ M $, with CR distribution $\mathcal D^{10}\subset T^\mathbb C M $, is called {\it totally nondegenerate of depth $\mu $} if:(a) the complex tangent space $ T^\mathbb CM $ is generated by all complex vector fields that might be determined by iterated Lie brackets between at most $\mu $ fields in $\mathcal D^{10}+\overline {\mathcal D^{10}} $;(b) for each integer $2\leq k\leq\mu-1$, the families of all vector fields that might be determined by iterated Lie brackets between at most $ k $ fields in $\mathcal D^{10}+\overline {\mathcal D^{10}} $ generate regular complex distributions;(c) the ranks of the distributions in (b) have the {\it maximal values} that can be obtained amongst all CR manifolds of the same CR dimension and satisfying (a) and (b)--this maximality property is the {\it total nondegeneracy} condition. In this paper, we prove that, for any Tanaka symbol $\frak m=\frak m^{-\mu}+\ldots+\frak m^{-1} $ of a totally nondegenerate CR manifold of depth $\mu\geq 4$, the full Tanaka prolongation of $\frak m $ has trivial subspaces of degree $ k\geq 1$, ie it has the form $\frak m^{-\mu}+\ldots+\frak m^{-1}+\frak g^ 0$. This result has various consequences. For instance it implies that any (local) CR automorphism of a regular totally nondegenerate CR manifold is uniquely determined by its first order jet at a fixed point of the manifold. It also give also a complete proof of a conjecture by Beloshapka on the group of automorphisms of homogeneous totally nondegenerate CR manifolds.
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