Whenever all the owners and other persons having a vested interest in the lands abutting on any street, alley, or other public place, or any portion thereof, in any plat of state lands, lying outside the limits of any incorporated city or town, petition the department, the department may vacate any such tract, alley, or public place or part thereof and in such case all such streets, alleys, or other public places or portions thereof so vacated shall be platted, appraised, and sold or leased in the manner provided for the platting, appraisal, and sale or lease of similar lands. However, where the area vacated can be determined from the plat already filed it shall not be necessary to survey such area before platting the same. The owner or owners, or other persons having a vested interest in the lands abutting on any of the lots, blocks, or other parcels platted upon the lands embraced within any area vacated as provided in this section, shall have a preference right for the period of sixty days from the date of filing with the department such plat and the appraisal of such lots, blocks, or other parcels of land, to purchase the same at the appraised value thereof.