(1) The department may provide a distinctive license plate for all trailers owned or operated by the state, counties, municipalities, or school districts. Trailers owned or operated by the state, counties, municipalities, or school districts shall display such distinctive license plates when such license plates are issued or shall display undercover license plates when such license plates are issued under section 60-3,135.

(2) Any trailer owned or leased and used by any city or village of this state, any rural fire protection district, the Civil Air Patrol, any public school district, any county, the state, the United States Government, any entity formed pursuant to the Interlocal Cooperation Act, the Integrated Solid Waste Management Act, or the Joint Public Agency Act, or any municipal public body or authority used in operating a public passenger transportation system, and exempt from a distinct marking as provided in section 81-1021, may carry license plates the same design and size as are provided in subsection (1) of this section or undercover license plates issued under section 60-3,135.