Context

By acting naturally via $\mathsf{SL}_3$ on $\mathbb{RP}^2=\{[x:y:z]\}$ and by taking the induced action on the affine hyperplane $z=1$ (which we identify with $\mathbb{R}^2$), one can realize the corresponding Lie algebra $\mathfrak{sl}_3$ as a sub-algebra of the algebra $\mathfrak{X}(\mathbb{R}^2)$ of vector fields of $\mathbb{R}^2$: by carrying out this simple exercise, one ends up with 8 vector fields, 6 of which are linear, and two contain quadratic coefficients, namely $$ X_A=-xy\partial_x-y^2\partial_y\, ,\quad X_B=-x^2\partial_x-xy\partial_y\, , $$ that correspond to the matrices $$ A=\left(\begin{array}{ccc}0&0&0\\ 0&0&0\\0&1&0\end{array}\right)\,,\quad B=\left(\begin{array}{ccc}0&0&0\\ 0&0&0\\1&0&0\end{array}\right)\, , $$ respectively. Image below shows how to use the flow of $X\in\mathfrak{sl}_3$ to obtain a vector field on the plane $z=1$:

The 6 linear vector fields are of course complete, whereas the flows of $X_A$ and $X_B$ display a singularity at $t=-\tfrac{1}{y}$ and $t=-\tfrac{1}{x}$, respectively. This remark motivated the first question:

For dimensions reasons (as observed in the comments), for every $n\ge 1$ there is no embedding $$ \mathfrak{sl}_{n+1}\hookrightarrow\mathfrak{X}(\mathbb{R}^n)\, , $$ whose image is made of linear (allowing a constant) vector fields.

However for any $n\geq 1$ it is easy to find an embedding $$ i_{\textrm{nat}}:\mathfrak{sl}_{n+1}\hookrightarrow\mathfrak{X}(\mathbb{R}^n)\, , $$ which we may call natural, because it is induced by the natural action of $\mathsf{SL}_{n+1}$ on $\mathbb{RP}^n$ by the same construction as above, one may wonder the following (initially "Question 2"):

Question

Given an arbitrary embedding $$ i :\mathfrak{sl}_{n+1}\hookrightarrow\mathfrak{X}(\mathbb{R}^n)\, , $$ it is true that $i$ must necessarily be equivalent to $i_{\textrm{nat}}$? If not, which conditions must $i$ satisfy, to be such?

In other words, one has a collection $X_1,X_2,\ldots, X_{d(n)}$, where $d(n)=(n+1)^2-1=\dim \mathsf{SL}_{n+1}$ of vector fields on $\mathbb{R}^n$, that commute according to the commutation relations of $\mathsf{SL}_{n+1}$ and the question can be recast as follows: is there a diffeomorphism of $\mathbb{R}^n$ that transforms the aforementioned vector fields into the images via $i_{\textrm{nat}}$ of a suitable set of generators of $\mathfrak{sl}_{n+1}$? What kind of obstruction (if any) one should expect?