Does there exist an undecidable problem such that the answer is YES for exactly one input to a UTM, and NO for all others?
For every Turing machine over some language L, there is a set of words that it will halt and accept on and a set of words it will halt and reject on, if it does halt.

If $|L| = 1 $ then the Turing machine will accept on exactly one input.

Given an arbitrary M and some input, we cannot decide whether the program will halt; halting problem. However, since M is not arbitrary, it’s fixed,this is not the Halting problem. I believe an algorithm would only need to check if the input is equal to the first element in L.