Conn. Gen. Stat. § 9-446

(b) If two or more candidates obtain the same number of votes at a primary held to nominate candidates for a municipal office, other than the municipal office of state senator or state representative, or to elect members of a town committee, or if two or more slates of candidates obtain the same number of votes at a primary held for justices of the peace, and a tie vote thereby occurs, any of such candidates, or the town chairman of the political party, may apply for a recanvass of the returns in the manner provided in section 9-445 . If no such application is made, or if any such recanvass results in a tie vote, such primary shall stand adjourned for three weeks at the same hour at which the first primary was held. Ballots of the same form and description as described in section 9-437 shall be used in the primary on such adjourned day, and the primary shall be conducted in the same manner as on the first day, except that the votes shall be cast for such office only. Ballots for the primary on such adjourned day shall be provided forthwith by the registrars of voters of the municipality wherein such primary stands adjourned, and the clerk of the municipality shall furnish the Secretary of the State with an accurate list of all candidates to be voted for at such adjourned primary. The clerk of the municipality wherein such primary so stands adjourned shall, at least three days prior to the day of such adjourned primary, give notice of the day, hours, place and purpose thereof by publishing such notice in a newspaper published in such municipality or having a circulation therein. No such primary shall be held if prior to such primary all but one of the candidates for such office die, withdraw their names or for any reason become disqualified to hold such office, and, in such event, the remaining candidate shall be deemed to be lawfully voted upon as the candidate for such office. No withdrawal shall be valid until the candidate who has withdrawn has filed a letter of withdrawal signed by such candidate with the municipal clerk. When such a primary is required to be held under the provisions of this section and prior to such primary all but one of the candidates for such office die, withdraw their names or for any reason become disqualified to hold such office, the Secretary of the State shall forthwith notify the municipal clerk of such fact, and shall forthwith direct the clerk that such primary shall not be held. In the case of a multiple-opening office only the names of those candidates whose votes are equal shall be placed on the ballot of the adjourned primary. If such second primary results in a tie vote, the registrar, in the presence of not fewer than three disinterested persons, and after notification to the candidates obtaining the same number of votes and the chairperson of the town committee of the party holding the primary of the time when and the place where such tie vote is to be dissolved, shall dissolve such tie vote by lot. The registrar shall execute a certificate attesting to the result of the dissolution of such tie vote, and the person so certified or the slate so certified as having been chosen by lot shall be deemed to have received a plurality of the votes cast and shall be deemed to have been chosen as the nominee of such party to such office.