An attempt to differential Galois theory of second order polynomial system and solvable subgroup of Möbius transformations

By introducing the conception “relativistic differential Galois group” for the second order polynomial systems, we establish the relation between the conformal relativistic differential Galois group and the subgroup of Möbius transformations, and prove that the system is integrable in the sense of Liouville if its conformal relativistic differential Galois group is solvable with a derived length at most 2. Some omissions on the structures of solvable subgroups of Möbius transformations at the first author’s article published in this journal in 1996 are refreshed in this paper.