Alex,
At first the misspelled name threw me off. I went through a lot of resources before I came upon the fact that the name was Brandan and not Brandau!

This is your man:
Gustavus R. Brandan
Surgeon
9th tennessee Cavalry, US Volunteers

He is also listed as surgeon in the 11th Tenn Cav.

At the top of this page is an ad. You can obtain a service record for any soldier here. If a record exists this is where to obtain it courtesy of Mr. Alan Pitts.

If you have an interest in the 2 units may I suggest you check this:

http://www.tngennet.org/civilwar/usacav/usa9cav.html

and:

http://www.tngennet.org/civilwar/usacav/usa11cav.html

These units were made up men from East Tennessee which was staunchly Union. As a matter of fact the county where I live temporarily, Johnson County, Tennessee was almost all Union. The only Confederate regular unit to be raised here was Co. A of the 6th North Carolina Cavalry. I am a Virginian and my ancestors were all Confederate. The two units you are interested in, fought in East Tennessee, Kentucky, Southwest Virginia and Western North Carolina.