Let $A = k[x_1,\ldots, x_n]$ be a polynomial algebra over a field of characteristic $p$.
Let $Der_k(A)$ denote the Lie algebra of derivations of $A$.
As we know, the Jacobian conjecture provides a criterion for a $n$-tuple of polynomials to be a set of variables.
We can ask the following question : what are (provable) criterions which tell whether the $n$-tuple of derivations $(D_1,\ldots, D_n)$ is the $n$-tuple of partial derivations $\partial_{y_1},\ldots, \partial_{y_n}$ in some polynomial basis of $A$?
${\bf Conjecture.}$ There is a family of variable derivations $D_1,\ldots, D_{pn^n}$ and the set $R$ of $pn^n$ relations of the form $$[D_a, D_b]= D_aD_b - D_bD_a = D_c, D_a + D_b = D_c, D_a = 0, D_a\not= 0, D_a = x_iD_b, $$$$1\leq a, b, c\leq pn^n$$ such that
$$R((D_1,\ldots, D_{pn^n})) = true \Leftrightarrow (D_1,\ldots, D_n) = (\partial_{y_1},\ldots, \partial_{y_n})\text{ in some polynomial basis }y_1,\ldots, y_n\text{ of }A.$$
${\bf Conjecture ((p, n) = (2, 2)).}$ In the particular case $(p, n) = (2, 2)$, $A = k[x_1, x_2]$ is it true that
$$\left\{(D_1, D_2, D_3)\in\text{Der}_k(A)\left\vert\begin{array}{cl} D_1, D_2, D_3\not= 0,\hfill\\ [D_1, D_2] = 0,\hfill\hfill\\ [D_1, D_3] = D_2,\hfill\hfill\\ [D_2, D_3] = 0,\hfill\\ [D_j,[D_j, x_iD_j]] = 0, 1\leq i\leq 2, 1\leq j\leq 3\end{array}\right.\right\}\Leftrightarrow$$$$\Leftrightarrow (D_1, D_2, D_3) = (\partial_{y_1}, \partial_{y_2}, y_1\partial_{y_2})\text{ in some polynomial basis }y_1, y_2\text{ of }A?$$
${\bf Motivation.}$ In the case $n = 2$ (and similarly in the general case $n\geq 1$), $k$ - field of characteristic zero the following analogue can be proved
${\bf Theorem.}$ $$\left\{(D_1, D_2, D_3)\in\text{Der}_k(A)\left\vert\begin{array}{cl} D_1, D_2, D_3\not= 0,\hfill\\ [D_1, D_2] = 0,\hfill\hfill\\ [D_1, D_3] = D_2,\hfill\hfill\\ [D_2, D_3] = 0,\hfill\\ \exists N\geq 1 :\\ \underbrace{[D_j, [D_j, \ldots [D_j, x_iD_j]\ldots]]}_{N\text{ times}} = 0, 1\leq i\leq 2, 1\leq j\leq 3\end{array}\right.\right\}\Leftrightarrow$$$$\Leftrightarrow (D_1, D_2, D_3) = (\partial_{y_1}, \partial_{y_2}, y_1\partial_{y_2})\text{ in some polynomial basis }y_1, y_2\text{ of }A.$$
