Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

First, I will review the classification of locally diffeomorphic finite-dimensional Lie algebras of analytic vector fields on the plane, accomplished by Sophus Lie, following the modern approach by Artermio Gonzalez-Lopez, Niki Kamran, and Peter J. Olver, who also clarified certain issues in the initial classification. I will study which Lie algebras of the classification are diffeomorphic to Lie subalgebras of others, as well as other relevant properties. Then, I will determine the subclass of Lie algebras that are locally Hamiltonian relative to a symplectic structure. Finally, I will explain how to use the classification to study relevant types of Hamiltonian systems on the plane and other related results.