Whenever the Office of President of the United States becomes vacant, whether by resignation, impeachment, ill health, whatever, the reason is immaterial.
Naturally, the Vice President takes over the presidency. No problem: sadly, it's happened before.
One of the new President's necessary acts is to select a new Vice-President, to fill out the remainder of the term before the next election. The selection must be approved by both the House and the Senate. However, the Senate is evenly split at 50 - 50.
Now there have been tie votes in the Senate, and they are resolved by the President of the Senate casting a deciding vote if he/she so chooses.
But, Constitutionally, the Vice-President is the President of the Senate! And there isn't one yet! The tie vote is to confirm the selection of a Vice-President!
So, in the absence of a Vice-President, how is a tie in a Senate vote decided?