The security deposited in compliance with the Motor Vehicle Safety Responsibility Act shall be placed by the department in the custody of the State Treasurer and shall be applicable only to the payment of a judgment or judgments rendered against the person or persons on whose behalf the deposit was made, for damages arising out of the accident in question in an action at law begun not later than two years after the date of such accident or within two years after the date of deposit of any security under subdivision (4) of section 60-511. The deposit or any balance of the deposit shall be returned to the depositor or his or her personal representative (1) when evidence satisfactory to the department has been filed with the department that there has been a release from liability, a final adjudication of nonliability, a supersedeas bond to insure payment of judgment filed and approved as set forth in subdivision (2) of section 60-511, a warrant for confession of judgment, or a duly acknowledged agreement, in accordance with subdivision (4) of section 60-510, or (2) whenever, after the expiration of two years from the date of the accident, or within two years after the date of deposit of any security under subdivision (4) of section 60-511, the department shall be given reasonable evidence that there is no such action pending and no judgment rendered in such action left unpaid.