If the amount deducted from the salary of a deceased policeman for annuity purposes after the effective date has not been paid to him, and in the case of a deceased married male policeman to him and his widow together, in form of annuity before the death of the last of such persons, the difference between such amount and the amount paid as annuity or annuities, without interest upon either such amount, shall be refunded to a surviving widow not entitled to receive an annuity under this Article. If there is no widow, the refund shall be paid to the children of the policeman, in equal parts, unless the policeman shall direct in writing, sworn to before an officer authorized to administer oaths in this State, and filed with the board before his death, that any such amount shall be refunded to the widow or to any one or more of the children, either or both. If any child is less than age 18, such part or all of any such amount equal to the sum necessary to pay children’s annuities under this Article for each such child shall not be refunded, but shall be transferred to the Child’s Annuity Reserve. If there are no children, the refund shall be payable to the executor or administrator of the policeman’s estate. If there is no executor or administrator, the refund may be applied toward the payment of burial expenses of the policeman, and any remainder shall be paid to his heirs who are living immediately after the death of the widow according to the law pertaining to the estates of deceased persons.