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Transitive local Lie algebras of vector fields can be easily constructed fromdilations of $\mathbb{R}^n$ associating with coordinates positive weights (giveme a sequence of $n$ positive integers and I will give you a transitivenilpotent Lie algebra of vector fields on $\mathbb{R}^n$). It is interestingthat all transitive nilpotent local Lie algebra of vector fields can beobtained as subalgebras of nilpotent algebras of this kind. Starting with agraded nilpotent Lie algebra one constructs graded parts of its Tanakaprolongations inductively as `derivations of degree 0, 1, etc. Of course,vector fields of weight $k$ with respect to the dilation define automaticallyderivations of weight $k$, so the Tanaka prolongation is in this case neverfinite. Are they all such derivations given by vector fields or there areadditional `strange'ones? We answer this question. Except for special cases,derivations of degree 0 are given by vector fields of degree 0 and the Tanakaprolongation recovers the whole algebra of polynomial vectors defined by thedilation. However, in some particular cases of dilations we can find `strange'derivations which we describe in detail

Transitive nilpotent Lie algebras of vector fields and their Tanaka prolongations