N.Y. Comp. Codes R. & Regs. tit. 22, § 1250.15 – Calendar preference; calendar notice; oral argument; post-argument submissions.

(1) By letter. A party seeking and entitled by law to a preference in the hearing of an appeal shall provide prompt notice by letter to the court setting forth the basis for such preference.
(2) By motion. A party not entitled to a preference by law may move for a calendar preference for good cause shown.
(b) Calendar notice.
Notification that a cause has been placed on the calendar shall be published on the court’s website. The court may also arrange for publication of such notice in a daily law journal or other newspaper or periodical regularly published within the Judicial Department.
(c) Oral argument.
(1) Oral argument generally. Oral argument shall be permitted unless proscribed by court rule or, in a particular cause, by the court in its discretion. Parties who do not file a brief on appeal shall not be permitted to argue a cause.
(2) Oral argument by permission. Where oral argument is proscribed by rule, a party may seek leave of the court therefor by filing of a letter application, on notice to all parties, or by motion where required by the court, within seven days of the filing of the respondent’s brief. The application or motion shall specify the reasons why oral argument is appropriate and the amount of time requested.
(3) Failure to request oral argument. In the event that any party's main brief shall fail to set forth the appropriate notations indicating that the cause is to be argued and the time required for argument, the cause will be deemed to have been submitted without oral argument by that party.
(4) Failure to appear for oral argument. Where counsel or a self-represented litigant fails to appear timely for oral argument, the matter shall be deemed to have been submitted without oral argument by that party.
(5) Rebuttal. Prior to beginning argument, the appellant may orally request permission to reserve a specific number of minutes for rebuttal in the First and Third Judicial Departments. The time reserved shall be subtracted from the total time assigned to the appellant. The respondent may not request permission to reserve time for sur-rebuttal.
(d) Post-argument submissions.
Post-argument submissions are discouraged, and may be made only with leave of the court.

(a) Calendar preference.