A married woman is a joint guardian of her children with her husband, with equal powers, rights and duties in regard to them. Upon the death of either father or mother, the surviving parent, whether of full age or a minor, of a child likely to be born, or of any living child under the age of eighteen years and unmarried, may, by deed or last will, duly executed, dispose of the custody and tuition of such child during its minority or for any less time, to any person or persons. Such surviving parent may appoint a guardian or guardians of the person and of the property of the infant and in making such appointment shall not be limited to the appointment of the same person or persons in both capacities. Either the father or mother may in the life-time of them both, by last will duly executed, appoint the other the guardian of the person and property of such child, during its minority. Either the father or mother may in the life-time of them both by last will duly executed, and with the written consent of the other duly acknowledged, appoint the other and a third person to be the guardians of the person and property of such child during its minority, and in making such appointment shall not be limited to the appointment of the same person or persons in both capacities. Such consent must have as part thereof a sworn statement that the consenting parent in so consenting, is motivated solely by the welfare of the child or children, the guardianship of whom is the subject of such consent, and that such consenting parent has not received and will not receive any consideration for such consent, and such consent may be revoked by such consenting parent at any time prior to the death of the other, by filing in the office of the county clerk of the county in which said other then resides, a written revocation of such consent, subscribed and acknowledged by the person so revoking, with proof of service of a copy thereof on such other parent in the manner provided for service of a summons. An appointment of a guardian of the person and property of an infant made by duly executed last will of his father or mother shall be valid and effective if at the time the will is admitted to probate the other parent shall have died or the surviving parent be an adjudicated incompetent. If both parents die under circumstances which render it difficult or impossible to determine which of them died first and both of them left last wills appointing the same person as guardian, the appointment shall be valid and effective. If both parents die under circumstances which render it difficult or impossible to determine which of them died first, leaving last wills appointing different persons as guardians, the surrogate’s court shall determine which of the appointments will best serve the welfare of the child and issue letters of guardianship accordingly. If at any time during the minority of the infant the surviving parent becomes competent to serve as guardian, he may apply to the court which issued letters of guardianship to the guardian appointed by will for a decree revoking such letters and the court shall on such application make such order or decree as justice requires. A person appointed guardian in pursuance of this section shall not exercise the power of authority thereof unless such will is admitted to probate, or such deed executed and recorded as provided by SCPA 1710.