Local extension of the translation group of a plane to a locally doubly transitive transformation Lie group of the same plane

In this paper, we examine the problem of finding all locally doubly transitive extensions of the translation group of a two-dimensional space. This problem is reduced to the search for finding Lie algebras of locally doubly transitive extensions of the translation group. The basis operators of such Lie algebras are found from solutions of systems of second-order differential equations. We prove that the matrices of these systems commute with each other and can be simplified by reduction to the Jordan form. From the solutions of systems of differential equations, the Lie algebras of all locally doubly transitive extensions of the translation group of the plane are obtained. Using the exponential mapping, we calculate locally doubly transitive Lie transformation groups.