Does President Biden have the authority to supply Ukraine with nuclear weapons?

President Biden announced that he will not send combat troops to fight in Ukraine.

That he does not want to have US armed men in the line of fire means that he also would not want to put US civilians in the line of fire.

Yet responding to a nuclear attack (of any kind) by Russia on Ukraine can be prevented with certainty only with a promise of a retaliatory attack that assures mutual destruction (where "mutual" here is Ukraine-Russia-US, perhaps in that order).

President Biden has, however, sent weapons. He didn't send the most sophisticated weapons in the US arsenal. He did not send Ukraine, for example, long range missiles that can reach Moscow.

What is the limit of the weapons that President Biden can authorize sending to Ukraine? Notably, does President Biden have the authority to send nuclear weapons to Ukraine, for those weapons to be under the control of President Zelenskyy?

The purpose of this question is to solve the following (unanswerable, or at least very difficult) question:

What can President Biden do to convince President Putin not to go nuclear?

• Responding with either "deploying a low-yield tactical nuke for show" (source: Matthew Kroenig at the Atlantic Council, a think tank) or "a conventional US strike on Russian forces" (source: Andreas Kluth, Bloomberg Opinion columnist) would lead to
  Armageddon.