Description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated

The main results of this article are: 1) the classification of those finite-dimensional filtered Lie algebras over fields of prime characteristic whose associated graded Lie algebras are Lie algebras of Cartan type; 2) a description of the infinite-dimensional primitive transitive subalgebras of Lie algebras of linear differential operators with coefficients in a subring of the ring of formal power series over a field of characteristic zero.