In order to study the infinitesimal symmetries of certain PDEs, Lie attempted to classify the possible infinitesimal actions of complex and real Lie groups acting on manifolds of dimension at most 3. Although incomplete in dimension 3, this classification highlighted several interesting problems; one of them being the classification of Lie algebras of so-called contact vector fields in the first-order jet space of a complex or real variable. While the complex case was addressed by Lie at the end of the 19th century, the real case was only made explicit by Doubrov-Komrakov in 1999, notably building on the work of Tanaka.

After presenting some interesting properties of these well-known contact vector fields, I will introduce the geometric idea brought by Tanaka's work, how it provides a solution to the problem in the real case through weighted multivariable Taylor truncations, and how this construction of approximations turns out to be exact.