On the local extension of the group of parallel translations in three-dimensional space. II

This article solves the problem of local extension of the group of parallel translations of a three-dimensional space to a locally bounded exactly doubly transitive group of Lie transformations of the same space. Locally bounded exactly twice transitivity means the existence of a unique transformation that takes an arbitrary pair of non-coinciding points from some open neighborhood to almost any pair of points from the same neighborhood. The problem posed is solved for four cases related to Jordan forms of third-order matrices. Using these Jordan matrices, systems of linear differential equations are written, the solutions of which lead to the basis operators of a six-dimensional linear space. Requiring that the commutators of the basis operators be closed, we find Lie algebras. By checking the condition of bounded exactly twice transitivity, we obtain the Lie algebras of the required Lie transformation groups. At the end of the paper it is proved that these Lie algebras are either solvable or representable as a direct sum of a solvable ideal and a subalgebra isomorphic to $sl(2,R)$. In this case, solvable Lie algebras are decomposed into the direct sum of a nilpotent ideal and a solvable subalgebra. Finally, the isomorphism of some above found Lie algebras is established.