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Second order anti-holonomic jets as anti-symmetric parts of second order semi-holonomic jets are introduced. The anti-holonomic nature of the Lie bracket is shown. A general result on universality of the Lie bracket is proved.

Let $F_1$ be a natural bundle of order $r_1$; a basis of the $s$-th order differential operators of $F_1$ with values in $r_2$-th order bundles is an operator $D$ of that type such that any other one is obtained by composing $D$ with a suitable zero-order operator. In this article a basis is found in the following two cases: for $F_1=\text{semi}{F}^{r_1}$ (semi-holonomic $r_1$-th order frame bundle), $s=0$, $r_2<r_1$ and $F_1=F^1$ ($1$-st order frame bundle), $r_2\le s$. The author uses here the so-called method of orbit reduction which provides one with a criterion for checking...