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New York Shall contain the word "corporation", "incorporated" or "limited", or an abbreviation of one of such words; there is also a long list of words a business corporation is not allowed to use without additional approval from other agencies including "board of trade", "state police", "urban development", "chamber of commerce", "state trooper", "urban relocation", "community renewal", "tenant relocation", "acceptance", "endowment", "loan", "annuity", "fidelity", "mortgage", "assurance", "finance", "savings" and many others New York State Consolidated Laws, Business Corporations Law § 301; Not-For-Profit Corporations Law, § 301
On the other hand, if you are not responsible for its tax, then you should declare the equipment in Part III of the Business Property Statement (Equipment belonging to others). Where equipment is declared in Part III of the Business Property Statement, the Assessor will also send a Notice to File to the person reported as the equipment's actual owner.