On the Local Extension of the Group of Parallel Translations in Three-Dimensional Space. II

We solve the problem oflocal extension of the group of parallel translationsin three-dimensional space to a locally bounded exactly double transitive Lie groupof transformations of the same space.Locally bounded exact double transitivity impliesthe existence of the unique transformation that sends an arbitrary pair of noncoinciding pointsin an open neighborhood to an almost any pair ofpoints in the same neighborhood.The problem is solved in four casesconnected with the Jordan forms of matrices of third order.Using these matrices, we write some systemsof linear differential equations, whose solutions lead tothe basis operators of a six-dimensional vector space.We find some Lie algebras by imposing the conditions of commutator closedness of the basis operators.We also obtain the Lie algebras of desired Lie groups of transformationsby verifying the condition ofbounded exact double transitivity.These Lie algebras are shown to be eithersolvable or represented by the direct sumof a solvable ideal and a subalgebra isomorphic to $ sl(2,R) $ .Furthermore, the solvable Lie algebras are decomposedinto the direct sum of a nilpotent ideal and a solvable subalgebra.Finally, we establish some isomorphisms ofa few Lie algebras found above.