  Let (E,D,P) be a flat vector bundle with a parabolic structure over a
punctured Riemann surface, (M,g). We consider a deformation of the harmonic
metric equation which we call the Poisson metric equation. This equation arises
naturally as the dimension reduction of the Hermitian-Yang-Mills equation for
holomorphic vector bundles on K3 surfaces in the large complex structure limit.
We define a notion of slope stability, and show that if the flat connection D
has regular singularities, and the Riemannian metric g has finite volume then E
admits a Poisson metric with asymptotics determined by the parabolic structure
if and only if (E,D,P) is slope polystable.
