Logarithmic connections with vanishing pp-curvature

We examine logarithmic connections with vanishing pp-curvature on smooth curves by studying their kernels, describing them in terms of formal local decomposition. With the goal of providing tools to work explicitly with such connections, we apply our results in the case of connections on bundles of rank 2 on the projective line, classifying such connections in terms of rational functions on the projective line with prescribed ramification.