The Historical Data Systems, Inc. database has Eli Branie as Pvt C 2 Cherokee Cav CSA
They do not have him listed as Branic. There are no other Branies listed as serving for the CSA, and no Branics (however, there are 3 Branicks from VA).
The National Park Service's "Civil war Soldiers and Sailors System has Eli Branie as Pvt C 2 Cherokee Mtd Vols CSA (this would probably the correct unit designation). It also lists a David Branie of 30 OH Inf (USA) and W.W. Branie of Hilliard's Legion AL Vols CSA. Only on Branic- Benjamin of 16 IL Inf.
I hope this information helps.