Suppose I have a universal Turing Machine (UTM) which accepts some input in binary. Is there a computational problem such that the answer to the problem is YES (accepting) for exactly one input (and hence NO for all others), but determining which input is the accepting one is undecidable? Or is there a fundamental reason this cannot happen? Or is determining if this problem exists undecidable?
I assume this is equivalent to asking: is there a property of TMs such that this property is true for exactly one TM and not true for all others such that determining which TM this is true for is undecidable?