Conn. Gen. Stat. § 9-224a: No election if only one candidate in special election to fill vacancy.

At any special election called to fill a vacancy in a state, district, or municipal office if there appears on the ballot the name of only one candidate, and no person has registered as a write-in candidate as provided in section 9-224b, the special election shall not be held. In the case of a municipal office, the sole candidate shall be declared elected by the municipal clerk and, in the case of a state or district office, by the Secretary of the State.