In any estate in which there is a will, on the death, resignation or removal of the sole surviving executor of any such will or of an administrator with the will annexed, administration of the estate of the testator, not already administered, may be granted with the will annexed; and in any estate in which there is no will, on the death, resignation or removal of the administrator of such estate, administration of the estate, not already administered, may be granted, to such person as the court shall find it proper to appoint; and no executor or administrator of an executor or administrator shall have authority as such to administer the estate of the first testator or the first intestate.