On the structure of transitively differential algebras

We study finite-dimensional Lie algebras of polynomial vector fields in $n$ variables that contain the vector fields ${\partial}/{\partial x_i} \; (i=1,\ldots, n)$ and $x_1{\partial}/{\partial x_1}+ \dots + x_n{\partial}/{\partial x_n}$. We derive some general results on the structure of such Lie algebras, and provide the complete classification in the cases $n=2$ and $n=3$. Finally we describe a certain construction in high dimensions.