Is there a single word that means "under the table"? I am looking for a single word that conveys a knowing, sly, violation of law or ethics — like an under-the-table or off-the-books payment.
If you're looking for something that is not known about to others, then surreptitiously.
Three possible adjectives are illicit, illegal and underhanded.
Kept or done in secret, often in order to conceal an illicit or improper purpose.
covert - Not openly practiced, avowed, engaged in, accumulated, or shown. thefreedictionary
Sub rosa, sometimes spelled "sub-rosa", means "under the table". It is a borrowed word, from Latin, but used in English language newspapers and elsewhere. It may be used as an adverb or an adjective. Synonyms are covertly and "behind the scenes".
In italian, for example, we have the one below (of course, the English version, which exists as well):
under the counter (or table) - (with reference to goods bought or sold) surreptitiously and typically illegally: certain labs have been peddling this drug under the counter. -|- [as adj.] an under-the-counter deal.
Some companies may have significant amounts of off-balance sheet assets and liabilities. For example, financial institutions often offer asset management or brokerage services to their clients. The assets in question (often securities) usually belong to the individual clients directly or in trust, while the company may provide management, depository or other services to the client. The company itself has no direct claim to the assets, and usually has some basic fiduciary duties with respect to the client. Financial institutions may report off-balance sheet items in their accounting statements formally, and may also refer to "assets under management," a figure that may include on and off-balance sheet items.
Mitch's answer is pretty good but the wrong part of speech. You want surreptitious.
A less formal (and thus possibly more appropriate depending on your use) term would be shady.