Metric extension problems for the static vacuum Einstein equations with prescribed geometric data on an inner boundary 2-surface arise in the context of Bartnik's proposal of a quasi-local mass in general relativity. We make the simplifying assumption of axisymmetry of both the Bartnik data and the static extensions. The Weyl-Papapetrou formulation of the static axisymmetric vacuum Einstein equations results in a free elliptic boundary value problem, which we propose to solve by coupling it to a geometric flow for the boundary surface. We study this new flow numerically and find axisymmetric static extensions for axisymmetric Bartnik data in many situations, including near round spheres in spatial Schwarzschild of positive mass. This is joint work with Carla Cederbaum and Markus Strehlau.