It seems that all thre of these men Soloman Brandenburg, Joshua Hicks and William Sandback which are mentioned as being a part of "Brandenbirch Cavalry" are listed in Bryan Howerton muster rolls of the Desha County Battalion of Arkansas Militia;

"A copy of a letter dated Laconia, (AR) Nov 28, 1861, attached to the roll is addressed to Gov. H.M. Rector, Commander in Chief, Arkansas Militia, b T. C. Fluornoy, Col. C.D.C.M. and it shows that the Court of Desha County, Ark in accordance with an ordinance of the State Convention passed in June 1861, organized the militia of the county according to the following plan: The county was divided into military districts (believed to be 13) and a company ordered to be organized in each district. This was done in July 1861 and all the officers of the battalion were commissioned from the County Court. If the emergency arises, which may require the services of the Desha County Battalion to the field, a call will be promptly responded to.
Roll endorsed,’Compared with original paper No. 3379, filed at Executive Office, Little Rock, Ark.’ "