33rd / 424th District Courts

Enabling statute is Tex. Gov't Code § 25.0291:
Burnet County has one statutory county court, the County Court at Law of Burnet County.
See statute on Lexis

Detailed provisions appear in Tex. Gov't Code § 25.0292.

The jurisdictional provision reads as follows:
25.0292
(a) Except as provided by Subsection (k), in addition to the jurisdiction provided by Section 25.0003 and other law, a county court at law in Burnet County has concurrent jurisdiction with the district court in family law cases and proceedings, including juvenile cases.

The general provisions conferring jurisdiction on county courts at law appear at Section 25.0003.