Before revision:  The new extended  system is similar to the one presented by Greengard et al. (2009) for which a fast direct solver was constructed by Zhang and Gillman (2018) but the corresponding fast direct solver is more efficient than the one for the original extended system.
After revision:  The new extended linear system is similar to the one presented by Greengard et al. (2009) for which a fast direct solver was constructed by Zhang and Gillman (2018) but the corresponding fast direct solver is more efficient than the one for the original extended system.