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. Another important feature of the new extended system is that it can easily handle problems that require specialized quadrature for weakly singular kernels  . Numerical results illustrate the improved performance of the fast direct solver for the new extended system when compared to the fast direct solver for the original extended system.
After 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 for the new extended linear system and that problems involving specialized quadrature for weakly singular kernels . Numerical results illustrate the improved performance of the fast direct solver for the new extended system when compared to the fast direct solver for the original extended system.