N.Y. Uniform Dist. Ct. Act § 1806
A person commencing an action upon a small claim under this article shall be deemed to have waived a trial by jury, but if said action shall be removed to a regular part of the court, the plaintiff shall have the same right to demand a trial by jury as if such action had originally been begun in such part. Any party to such action, other than the plaintiff, prior to the day upon which he is notified to appear or answer, may file with the court a demand for a trial by jury and his affidavit that there are issues of fact in the action requiring such a trial, specifying the same and stating that such trial is desired and intended in good faith. Such demand and affidavit shall be accompanied with the jury fee required by law and an undertaking in the sum of fifty dollars in such form as may be approved by the rules, payable to the other party or parties, conditioned upon the payment of any costs which may be entered against him in the said action or any appeal within thirty days after the entry thereof; or, in lieu of said undertaking, the sum of fifty dollars may be deposited with the clerk of the court and thereupon the clerk shall forthwith transmit such original papers or duly attested copies thereof as may be provided by the rules to the part of the court to which the action shall have been transferred and assigned and such part may require pleadings in such action as though it had been begun by the service of a summons. Such action may be considered a preferred cause of action. In any small claim which may have been transferred to another part of the court, the court may award costs up to twenty-five dollars to the plaintiff if he prevails.
N.Y. Uniform District Court § 1806