Eureka!

The reason there is no record for T. M. Anderson is that only the muster-in roll for the regiments companies exists in the files, that dated March 31, 1862.

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The index to Texas Confederate Pensions list:

Mrs T M Anderson in the Alabama, Texas and Virginia, Confederate Pensions, 1884-1958

Name: Mrs T M Anderson
Application Date: 7 Apr 1924
Application Place: Menard
Birth Year: abt 1853
Birth Place: Upshur, Texas
Age: 71
Spouse: Thos Anderson
Marriage Date: 8 May 1871
Marriage Place: Johnson, Texas
Spouse Death Date: 21 Jan 1911
Pension File Number: 40091
Application Type: Widow

……

Her application lists he served for two years as a Private in Company F, 19th Texas Cavalry
Application attested by Thomas Reese & Thomas E Cross, also of Company F, 19th Texas Cav.

Pension approved August 18, 1924, allowed from June 1, 1924. She died March 3, 1944

In correspondence with the Texas Pension Commissioners, the Adjutant General, War Department on June 9, 1924, writes:

The name of Thomas Anderson, has not been found on the muster-in roll of Co. F, 19th Texas Cavalry, C. S. A., dated March 31, 1862 (only roll on file)

Source Citation
Texas State Library and Archives Commission; Austin, Texas; Confederate Pension Applications, 1899-1975; Collection #: CPA16526; Roll #: 2371; Roll Description: Pension File Nos. 40081 to 40120, Application Years 1917 to 1924
Source Information
Ancestry.com. Alabama, Texas and Virginia, Confederate Pensions, 1884-1958 [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2010.
Original data: Texas, Confederate Pension Applications, 1899-1975. Vol. 1–646 & 1–283. Austin, Texas: Texas State Library and Archives Commission.