On the structure of graded transitive Lie algebras

We study finite-dimensional Lie algebras L of polynomial vector fields in n variables that contain the vector fields (partial derivative)/(partial derivativexi) (i = 1,...,n) and x(1)(partial derivative)/(partial derivativex1) +...+ x(n)(partial derivative)/(partial derivativexn). We show that the maximal ones always contain a semi-simple subalgebra (g) over bar, such that (partial derivative)/(partial derivativexi) is an element of (g) over bar (i=1,...,m) for an m with 1 less than or equal to m less than or equal to n. Moreover a maximal algebra has no trivial (g) over bar -modules in the space spanned by (partial derivative)/(partial derivativexi) (i = m + 1,... n). The possible algebras (g) over bar are described in detail, as well as all (g) over bar -modules that constitute such maximal L. The maximal algebras are described explicitly for n less than or equal to 3.