Minn. Stat. § 204D.20: NOMINATIONS; VACANCY.

Subdivision 1. Special primary.

Except as provided in subdivision 2, the candidates of the major political parties to fill a vacancy shall be nominated at a special primary. The candidate of each party who receives the highest number of votes at the special primary shall be nominated.

Subd. 2. No special primary; when.

No special primary shall be held to nominate candidates to fill a vacancy if only one individual from each major political party files as a candidate for that party's nomination. In that case, the individuals who have filed are nominated.

Subd. 3. Nominations by petition.

Candidates to fill a vacancy may also be nominated by petition under the conditions and in the manner provided by law for candidates filing by petition for like office at the state general election as far as practicable.