Poisson metrics on flat vector bundles over non-compact curves

Let (E, del, Pi) -> (M, g) be a flat vector bundle with a parabolic structure over a punctured Riemann surface. 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 del 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, del, Pi) is slope polystable.