Conn. Gen. Stat. § 50a-36: (Formerly Sec. 52-616). Stay of enforcement.

(a) If the judgment debtor shows the court that an appeal from the foreign judgment is pending or will be taken, or that a stay of execution has been granted, the court shall stay enforcement of the foreign judgment until the appeal is concluded, the time for appeal expires or the stay of execution expires or is vacated, upon proof that the judgment debtor has furnished the security for the satisfaction of the judgment required by the state in which it was rendered.

(b) If the judgment debtor shows the court any ground upon which enforcement of a judgment of a court of this state would be stayed, the court shall stay enforcement of the judgment for an appropriate period, upon requiring the same security for satisfaction of the judgment as is required in this state.