You've misread the entry: recorded " . . . later on the end of February, 1864 absent ' . . .in pursuit of a deserter,'"

He was assigned to hunt down a deserter.

These two websites should explain the unit designations:

http://www.tshaonline.org/handbook/online/articles/WW/fwo16.html
http://www.angelfire.com/tx3/RandysTexas/36thcav.html