In this paper we suggest new effective criteria for the density property. This enables us to
give a trivial proof of the original Anders\'en-Lempert result and to establish (almost free of
charge) the algebraic density property for all linear algebraic groups whose connected
components are different from tori or $\C_+ $. As another application of this approach we
tackle the question (asked among others by F. Forstneri\v {c}) about the density of algebraic
vector fields on Euclidean space vanishing on a codimension 2 subvariety.

In this paper we suggest new effective criteria for the density property. This enables us to
give a trivial proof of the original Andersén-Lempert result and to establish (almost free of
charge) the algebraic density property for all linear algebraic groups whose connected
components are different from tori or $\C_+ $. As another application of this approach we
tackle the question (asked among others by F. Forstnerič) about the density of algebraic
vector fields on Euclidean space vanishing on a codimension 2 subvariety.