All claims and other obligations of the city which are payable out of solvent funds shall be paid by check; provided, that when the applicable fund is not solvent at the time payment is ordered, a warrant shall be issued. The public depository upon which such checks shall be drawn shall be the banking institution then under contract with the city to provide primary banking services. The mayor and the city treasurer, and such other city officers or employees as specifically designated in writing by the mayor, shall be the only officers or employees of the city authorized to sign checks.