Thanks Linda!

There is, indeed, a pension file for Robert Burnett with widow Jane Burnett. AA Civil War Memorial - Plaque#: C-69 ¹
(1) Robert Burnett, -- Co. C, 58th U.S.C. Infantry [from] Co. C. 6th Miss-Infantry (African-Descent). ¹
Soldier's Pension Application: Aug 12, 1887 (#619139; no Cert#; MS)
Widow's Pension App: (Jane Burnett) - Application: Sep 3 1893 (# 559550; Cert#: 372195; MS)

There are other Burnetts, who filed in MS:

(2) Robert Burnett [Co. F, 50th U.S.C. Infantry]
Soldier's Pension App: 1890 (#894186; no Cert#; MS)
Per 1890 Census: Enlisted: 1865; Discharged: March 1866; Service: 2 yrs, 8 mos., --- days. ²

(3) Robert Burnett [Co. D, 61st Regiment, U.S.Col Infantry]
,Co. D, 2nd W. Tenn Infancry A.D.]
Soldier's Pension App: 4 Sep 1890 (Both App# & Cert#; Ark)
Widow's Pension App: (Becky Burnett, Ark, 1913)

(4) John Burnett [Co. H, 11th Regiment, U.S.Col. Inf. ]
Soldier's Pension App: (MS, 1889)

There are also a few other Burnetts in surrounding states.
Check out the Soldiers and Sailors Site for histories of the regiments, OR
Our 'Military Forum' for more information on researching your Civil War Veteran.

Sources:
¹ "Soldiers & Sailors System." The American Civil War. National Park Service - U.S. Department of the Interior. 08-May-2006 [//cwar.nps.gov/civilwar].

² National Archives and Records Administration. Civil War Pension Index: General Index to Pension Files, 1861-1934. [database online] Provo, UT: Ancestry.com, 2000. Data indexed and imaged from National Archives and Records Administration. General Index to Pension Files, 1861-1934. T288, 544 rolls. Washington, D.C.: National Archives and Records Administration.

³ Ancestry.com. 1890 Veterans Schedules [database online]. Provo, Utah: MyFamily.com, Inc., 2005. Original data: United States. Special Schedules of the Eleventh Census (1890) Enumerating Union Veterans and Widows of Union Veterans of the Civil War. M123, 118 rolls. National Archives and Records Administration, Washington D.C.