after [Chitour-Jean-Long 2013]

A module of vector fields on an n-dimensional manifold is bracket generating, its nonholonomy degree r is, say, big in comparison to n (r >> n). Its desingularization is a regular distribution of the same rank living on a much much bigger manifold. (One lifts up module's generators one by one. Together they span a regular distribution having a unique small growth vector (sgrv). Meaning that the lifted generators behave freely in the Lie algebra sense up to length r, when they start to span the full tangent bundle of the lifted manifold.) For instance, a not regular rank 2 distribution having at the reference point the small growth vector [2,3,4,5,5,6] gets desingularized on a 23-dimensional manifold. The sgrv of its desingularization, one and the same at every point, is Lie-free up to length six: [2,3,5,8,14,23].

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Such desingularization plays a key role in a recent, important in the sub-Riemannian geometry, preprint by Hakavuori and Le Donne.