N.Y. Real Prop. Law § 428
Any allowed claim for indemnity shall be paid in the same manner as other claims against the county. In the city of New York a claim shall be passed upon and approved by the city register and by the corporation counsel of the city before payment is allowed. The rejection of a claim by the proper county officials (or in the city of New York by the city register and corporation counsel) shall not preclude the claimant from bringing an action to recover such claim. No claim or judgment on a claim for indemnity shall be binding on the county or on the county treasurer (in New York city the commissioner of finance) for an amount exceeding the amount credited to the assurance fund. If the amount credited to the assurance fund is insufficient to pay the claim or judgment in full, the unpaid balance shall bear interest at the legal rate and shall be paid out of the first moneys coming into said assurance fund or the county or city general fund after the assurance fund has been terminated pursuant to section four hundred twenty-six of this article. If any right of action against any person for damages for negligence or other cause, or under any covenant or contract of warranty or guaranty or otherwise, exists in favor of the person to whom indemnity is paid, the county treasurer (in New York city the commissioner of finance) shall be deemed to be subrogated to such right and may bring an action to recover thereunder. Any amounts recovered by the county treasurer (in New York city the commissioner of finance) under such an action shall be credited to the account of the assurance fund. Until the assurance fund provided as aforesaid shall have been exhausted, payment for any such losses or damages shall be made out of such fund.
N.Y. Real Prop. Law § 428